Sorry, but I see a problem with both options:
To do that you'd need to turn down the ends so the bearings can slide on, negating the use of 3/4" shaft.
Although, leaving one end @ 3/4" will work, as long as the sprocket is 5/8".
the bearings needed to do this would be an odd size &, therefore, expensive.
A third option is to stay with the 5/8" shaft & use a spacer/shim & a deeper keystock.
Come to think of it, the larger key may not be necessary as the spacer would only be 1/16" thick
I don't like the options I suggested either, but if he is set on using the existing jack-shaft bearing housing they are the best options I can think of.
Using spacers could work, but if he doesn't add the support bearing I mentioned, moving up to a 3/4" jack-shaft would be even more important. I would say the oversize key would be necessary, the driven or the jack-shaft's key-way could end up engaging only 1/32" of the key's height. That sounds like trouble just waiting to happen.
Do you know the ID of the bearing housing? Until the OP measures I can't know for sure. If it is 1 3/8" (likely) then the required bearings would be an odd size, but how do you know it isn't 1 5/8"? Although .75" ID by 1.375" OD ball bearings are uncommon, they can still be found. I purchased some from Blaz (go karts of Texas) a year or two ago, they were a little expensive, but nothing ridiculous.
I'm not at all seeing what you are saying about my first suggestion not working. Take a 3/4" keyed shaft, turn down one end to fit the bearings. Slide that end through the bearings, and install something on the small end sticking out of the bearings to secure it (ie. a locking collar). Then you have a 3/4" keyed shaft sticking out of the bearings that you can place the driven, a sprocket and a support bearing.
I believe that a 40 series setup should look almost exactly like
THIS (sprocket can go on either side). Sure it will cost you a bit more, and yes you will probably need to remove the engine (four bolts?), but I think that having it built solidly will be worth it. If you put in the extra effort now you will never have to mess with it again.
mdj89, the length of the jack-shaft is something that you are probably going to have to figure out on your own.