welcome to the forum :-)
If price is an issue, there's a couple of things that come to mind.
In a german woodworking forum I once found a copy of a (members only) plan from fine woodworking to rout chair seats, not 100% sure you can reach it though - try here
Maybe you can find the original article as well
However, this is more or less a template / jig dedicated for seats - but easy and effective.
If you really need a copy carver / router, then the one from Bill Hyltons book "Router magic" is surely an effective and inexpensive one to build.
Both versions would need some sanding for a clean finish, depending on the router bits you use.
I can't tell yet from own experience - both those devices are on the list of things to do next summer, when my shop has thawed to a useable extend again - probably with some modifications, too
Outside routers, an easy attempt to remove bigger amounts of (unwanted) wood, might be the chainsaw adapters Lancelot and Squire from King Arthurs Tools
, to be used on the angle grinder, possibly even both combined.
I have even seen the usage of an angle grinder on the lathe for quick wood removal on Youtube - but can't find it again.
However, a similar approach would be possible with the finewoodworking jig and the King Arthurs Adapters as well, I'd say, without bigger modifications. You'd be close to the final shape in no time, with just the need for some final sanding.
Let me know if you need more details