To Wondering Wanderer. Hope this helps. I've only just picked up this thread. I too am a predominantly solo walker/trekker. I rarely camp nowadays(at 73 I appreciate a LITTLE bit of luxury. Two years ago I did some of the TMB(hoping to return next year) and purposely booked in advance at refuges and gites. Walking solo does not mean being lonely and there are advantages. Being solo you can walk at YOUR pace. I find that trying to keep up with the rest of a group can be mentally as well as physically tiring. You can modify your route to suit YOURSELF without having to worry about what others are doing. I find that I invariably meet people along the way and often stop for a chat. This in itself is great; you can exchange all manner of information about the route ahead, the weather, refuges and other snippets. Then, when you arrive at a refuge, you meet more people and engage in social intercourse, sometimes with others you have met previously.
I know camping can be cheaper but as I think somebody else mentioned you can always camp near a refuge.
I am rather surprised that you said that your chosen site at Les Houches was closed. I too made my TMB attempt at the beginning of September and everything was still very much in full swing. Chamonix was bustling, as far as I know camp sites were open and the refuges were busy.
One of the things I enjoy about such treks is the camaradie and I say that as a confirmed solo trekker.
There is a handy gite in Les Houches - Gite Michel Fagot - which I used on my first night before embarking on the tour. It was still open on my return although I used a cheap hotel in Chamonix on my last night.
Happy walking