I have found vehicle rentals to be cheapest in Puerto Montt more so than in any other part of Patagonia, with the caveat that most Chilean vehicle rental agencies are not automatically cleared for use in Argentina unless one arranges it before hand making it clear that you only want a vehicle that can cross into Argentina. Prior to 2004/5 it was relatively easy, however new Argentine vehicular insurance regulations implemented in that period have made it more difficult for Chilean registered vehicles to cross into Argentina. Whereas most Argentine vehicles do have the necessary permits to cross into Chile. My most recent experience in 2017 (from over 10 long visits from 1983 onwards) in Puerto Montt, Bariloche, Punta Arenas, Balmaceda (for Coyhaique) was very positive in that Puerto Montt agencies provided good service i.e. cheap, reliable and hassle free.
Issues that may come up in vehicle rental:
adding the necessary permits for rental vehicle entry into Argentina can add a significant cost to the rental, can be time consuming, and best be arranged and cleared with the agency prior to arrival (suggest calling directly rather than just assuming they will get the email, otherwise you may be in a situation in which the only vehicle with the necessary paperwork for entry into Argentina might not be available.
Adding a different return city can be prohibitively expensive re renting/hiring in Puerto Montt and returning in Punta Arenas as in a 50-100% extra surcharge on the total rental cost. It is possible, but the cost, logistics might be a deal breaker. If for example the agency has to return the vehicle to lets say Puerto Montt from Punta Arenas, it might have to do so by ferry which isn't cheap.
In 2017 I rented a vehicle in the different cities and towns rather than driving point to point because my traveling companion would have required a visa to enter Argentina otherwise we would have done so i.e point to point. If you rent in Punta Arenas you will be limited to the Chilean road network between Puerto Natales and Chilean Tierra del Fuego.
You cannot drive from the terminus (Villa O'Higgins) of the Carretera Austral to Punta Arenas without either a-backtracking through Argentina or b-by taking the relatively new ferry from Puerto Yungay/Caleta Tortel to Puerto Natales and from there to Punta Arenas via paved road. I am frothing at the mouth in anticipation of someday trying this new ferry connection as it had recently opened at the time of my latest visit in 2017. It really is a deal changer as prior to this new ferry crossing, you could only get to Punta Arenas via land from Argentina or via the Puerto Montt Navimag vehicular ferry to Puerto Natales and from there to Punta Arenas. With this new ferry link you can now connect the entire Carretera Austral from the terminus at Villa O'Higgins backtracking only slightly to Puerto Yungay, from there to Puerto Natales and Punta Arenas totally within Chile. Something that was not possible prior to this new link, as you had to cross into Argentina. However, In order for this itinerary to work within the time constraints you have, you must lock in (as in reserve the departure date that works for you) the Puerto Yungay-Puerto Natales ferry crossing as it only runs once a week (need to confirm their May departures). Otherwise if it is full, or not departing at the time of your visit, you will have to head north (backtrack) from Villa O'Higgins and cross into Argentina in the vicinity of Cochrane via Paso Roballos.
The entire area is sublime, breathtaking scenery, virtually any specific route would be worthwhile. However the weather may play an important part in your enjoyment of the area. The western bits i.e. towards the archipelagic Chilean side are extremely rainy particularly after March. Per my personal constraints I have rarely been able to travel to Patagonia during the summer months i.e. Nov-Feb, so most of my experience would be germane to your intended month of departure. In 2017 I had Indian summer weather, brilliant blue skies, forests tinged in autumnal colors, but have also been socked in by constant 24 hour overcast rain in 2005, 2007. So, the short answer would be that Patagonian weather in May is usually very rainy, but you never can tell. The weather is much colder and more reliably good in the drier Argentine side than on the Chilean side. May might be too late for prime autumnal viewing particularly in the southernmost areas as they turn red earlier than Northern Patagonia. In my 2017 trip, I had gorgeous autumnal colors in Chilean Tierra del Fuego at the end of March. I would not expect that area to still be clad in fall colors by May. It really depends as shifting weather patterns have completely altered what was and is normal.
Is it safe?
I can't think of a safer place anywhere except any large temperate wilderness as in North America, Northern Scandinavia, New Zealand. Safety will be largely contingent on the weather i.e. heavy snowfall can occur at any time, extreme rain, long gravel roads in poor weather can be a difficult proposition even within the familiarity of ones home country. Never underestimate the road conditions, weather, driving times as daylight hours are shorter in May than in December, January.
What to expect?
Simply some of the most glorious, outrageously beautiful scenery in the world. If South Island New Zealand, Tasmania, British Columbia, Alaska, the Western United States, Norway, Iceland hold any fascination or interest, then you would be very pleased with Patagonia.
Low Season lodging
A lot of the infrastructure shuts down, tour agencies close for the winter, proprietors of smaller establishments do close for the winter. However with a vehicle you should be able to always find a place to stay. Having an entel sim card will allow for real time communication with local hoteliers. Chilean cell phone service is quite reliable even in Southern Tierra del Fuego. The cell phone towers are usually placed parallel to the Carretera Austral so you will generally have a signal within a few kms prior to entering a hamlet and a few kms after departing it.
Suggestions?
One of my favorite routes is the Rio Tranquilo valley route off of the Carretera Austral in the southern part of Lago Carrera. You will obtain a close and personal view over the Northern Patagonian Icefield, and has the added benefit of having a year road lodging option in Campo Alacaluf run by a genial German couple. I hope they are still around. Paso Reballos to Argentina is simply beautiful. The area between Puyuhuapi to Queulat National Park is memorable. In fact I have to stifle the impulse to describe everything as sublime or outrageously beautiful. Closer to Puerto Montt the Futaleufu area is another memorable area with clear fast running rivers, streams, lakes, waterfalls, etc. There are quite a few hot springs re. Puyuhuapi *pricey, amarillo *cheap, clean, road accessible, etc. View the effects of the devastation wrought by the Chaiten Volcano in 2008, etc. The Hornopiren and Pumalin area is .... you get the picture. The weather can both hamper and add to the experience. You can expect constant rain in the western areas, but if it chills by a few degrees you will have a white winter wonderland with amazing views.
Summary
May is usually rainy, but if it gets colder you will have wonderful views and driving conditions as it is less slippery. It is probably too late in the season to get much fall color, unless you spend more time on the Argentine side as in Nahuel Huapi National Park, Less lodging options but you should have no problems finding one when you need it. Camping any place. Plan the trip around the ferry crossings i.e. after reserving a specific day on the ferries, especially the ones further south i.e. Puerto Yungay, otherwise you will not be able to drive to Punta Arenas without backtracking through Argentina.