I walked the OLT in the second week of March this year. I used a MSR Hubba NX and my two friends shared my Nemo Galaxi 2P (a 2.5kg tent they split up between their two packs). Neither are four season tents (they have mesh inners) but this was not a concern for us. Even if the weather got really, really nasty, you always have the options of the huts. Even if the sleeping areas are full, there is normally a patch of floor in the common area that you could use in a pinch. We have never camped in really, really bad weather so this was our plan

The Hubba NX is a small tent but it was fine - most of my non-sleeping related gear has to go outside in the small vestibule (on the OLT I just stored my pack up at the hut each night). I am 5' 10" and I personally thought that if I was too much taller, its length may become an issue (one morning I found the end of my quilt was wet from pushing against the end of the tent onto the fly). My friends were also fine in the Galaxi and enjoyed the door each side and the big vestibules (they left all their gear at camp except for food - the possums at Windermere were huge and had no fear!).
I am pretty sure I saw a Black Wolf Mantis 1 or 2 man tent being used by someone on the OLT - there are new UL versions that are around your price range now. As well, someone elsewhere posted about the Naturehike series of tents which are knock-offs of MSR, Big Agnes and other UL designs from China and are very affordable although still may be an unknown quantity. They have a one man version that looks a bit like an Hubba on eBay although it remains to be seen how much time was spent on making it look like a premium tent and how much time they spent on really important things like durability. As you have some time this is something you may be able to monitor as other people report how they go.
Of course, if this is the only time you may use a really small and light tent for hiking, hiring one may be the cheapest option of all.
Finally, you mostly pitch tents on platforms on the OLT. While having a free standing tent makes it easier to do, don't let this stop you from buying a non free standing tent. I took someones suggestions of taking small cup hooks which I could screw into the boards at the right spot. This made tent set up was a snap (especially for someone that never pitched on platforms before) although you may need to take a few longer guy lines as well.