Well it is just me but I think the trans siberian is two amazing places for food and culture separated by days of endless travel through next to nothingness.
Have you considered more time in China? You could go spend months in China exploring all it has to offer.
Singapore to Hanoi overland is full of food, culture, and adventure and probably works out to be cheaper than the trans siberian.
Or you could do all of east asia overland. Fly into Tokyo, take the train to Kyoto, then boat to Busan, Korea, take the train up to Seoul and then another boat to China before ending in Beijing.
Or you could do Eurasia overland but go a little further south, taking in the silk road. Start in Istanbul, head east into Georgia, Armenia, or Azerbaijan, then take a boat across the Caspian to Kazakstan, then head down through the ancient towns of Khiva, Samarkand, and Bukhara in Uzbekistan before crossing into the alpine mountains of Kyrgicstan. Then into China's Xinjiang province. Skirt Tibet taking in Labrang Monestary in Gansu before heading to Xi'an and the terracotta warriors, and then a train to Beijing. I suppose that would take months but it would certainly take in more food and culture than the trans siberian.