I purchased one of these same "transmissions" from an eBay vendor and got it last Friday. My impressions of it are similar to yours. It will mount up to my Subaru and also my Honda GX31, and although I haven't accurately measured it, the drum seems to be the correct size. (Unfortunately the clutch drum on mine has over 1/16" of run out, so I'll have to deal with the seller on this - so no fun here either.)
I think there may be some potential for your application with a similar transmission though. Someone posted a picture this other transmission earlier in this thread. This other one appears to use two separate chains in tandem and is a bit shorter overall. These transmissions are available from several eBay vendors and can be identified by their having a cover bolt pattern with one bolt on the very rear of the case.
Mounting the sprocket on such a long unsupported shaft would still be certain to be a problem though. You could probably design a support bearing out near the sprocket that would solve this issue. (Perhaps mounting the bearing on a plate that was bolted to the engine somehow?)
Anyway, in your case of buying a "bicycle engine kit", the kit should have been completely designed and tested by the "manufacturer/vendor" before offering it sale - which it obviously wasn't. Selling something like this that has obviously just been thrown together and offered for sale as a workable kit reflects VERY poorly on their general business integrity.
Don't get me started...!
EDIT:
A general description of this "transmission":
This unit uses a single #25 chain with a 9 tooth clutch shaft sprocket and a 27 tooth output shaft sprocket. The chain has a fair bit of slop and there is no way to adjust it (although you could perhaps make an adjustable slipper if you needed to). The output shaft is supported with two bearings - one 6201 and one 6000 with about 1/2" between them on the inside. The output shaft appears to be simply pressed into the hub of the sprocket with no key or splines. The housing is rather lightweight. All of the bearings are loose in their bores and with lots of side play. The case is not sealed in any way.
In summery, this transmission is a medeocre, light duty, grease lubed only, 1:3 reduction chain drive inside a case.