no, because the trill ends with grace note d, eb, and then b-d, so you need to have 2 available to hit the d. In the time between my post above and now I tried it, and I'm better at the fingering I listed above (1-3, trill 3-4, then 2,3, 1-2) than trying to trill 3-5. but the edition I have (schenker) lists 3-5 for the trill.
I have played the first movement of this sonata before and i used the fingering 3 and 5 but that is because i have large hands...this was easier for me...if u want more technical work i would actually recommend working on some of the chopin etudes mayb even the revolutionary