Thanks for the prompt responses. Based on what you all provided, it looks like the better part of valor for me will be to use a round-over bit for the top and bottom of the table top. It never occurred to me that the thickness of the table top may not be uniformly 3/4" all the way around, and the consequences of that being the case had I used the bullnose bit. I'm thinking that I should be able to get the full-round effect by routing the top round-over via the bit's bearing, and the bottom round-over using the same bit, but using my trammel (circle jig), which should avoid the need for a flat surface for the bearing. The router is attached to the base for my router table, so there's lots of support for routing the upper round-over. The circle jig likewise provides firm support for the router.
Thanks again for the great analysis and advice.