I have been in Uganda/Rwanda for nearly two months and have never seen anyone fail to get a Gorilla permit, with most people turning up at either an agent or the authority office the day before. I think the information in the LP guide about how hard these permits are to get hold of is rather out of touch though I do appreciate that things may change toward the summer.
Transport to and from the park is a bit of a disgrace with tour groups charging $100 and no taxis willing to take you for less than $80. The problem isnt so much getting to the park gate, its getting to the trailhead after the 7am briefing as depending on what group of Gorillas you are selected to see some can be up to 15km away and the track is very poor.
Your best bet if you want to do it cheap is to get to the park entrance from the Ugandan border and then I would say that there is a 90% chance of you being able to get a ride to the trailhead PROVIDED that there are other people on the same trip as you. I cannot believe that anyone would just leave you there while they ride off into the sunset.
A bus to the border is about 300RWF and from there the park entrance is about 20km. I am sure that there will be a car willing to take you to the park for about $10 from here. This will be a lot cheaper than getting to the park from Musanze as ALL of the drivers here just charge $80 as they are in a league with the tour operators (i assume as none could be talked down).
Another option might just be to offer any local with a 4x4 some money in Musanze, I am sure a lot would welcome $30 odd if they are not too busy.
Before you do anything though, fInd out from your tour group if there are other people doing the same trek as you. If you find this out now ask for their contact details and you might be able to arrange to get transport together or jump in with them to split the $100 fare.
Hope this helps!