1. Get another estimate. Your old four-string neck should be worth more than that.
2. Swapping is a good idea. See if there's a musican union in your area. They will have a newsletter with want ads in it. Somebody might be interested.
3. I worked in a musical instrument business in the '60's. We did a lot of conversions. If you're good with wood try the following: a. From the 4th fret to the banjo rim on the deep D-string side. Remove any edging, mother-of-pearl etc. The with a very light touch flatten about 3/4' ind of the neck. A draw-knife is the right tools. b. Find an old hardwood furniture leg, or table top and miter box a stripe about 3/4 high x 1/2 x by the lenght of distance from your 4th fret to the rim. It should be flat on top and on the side that mates with your banjo neck. Rounded on the remaining sides to fit the look and feel of the instrument. The reason you start at the fourth fret is to give you room for a 5th fret nut. c. Buy a 5th tuning machine and install in into the existing neck. Not the new wood, which will be too thin. The tuning machine can be an inch away (or more) from the 5th string nut. Doesn't matter, the strings are plenty long. d. Don't worry about frets. You don't need them for a drone string. Add on 5th string capos (I use model railroad spikes) are available if you want. e. Glue into place. If you're nervous about tension, countersink small (wood screws and plug with wood from the base stock). Remember to drill pilot holes or the screws will shatter the stripe of wood.
Then glue. And finish. I suggest a contrasting finish rather than trying to match an old finish. Much easier.
This will work. But it takes some cohones to cut into an old instrument even of dubious value. Not counting finishing, figure two hours work.
Totally alternate method: Take a trashed guitar neck and mill it down. Lop of the top to tuning machines. This is more work. But it has a lot of possibilities. If you have a sander/shaper and some experience: figure 5 hours.