I didn't know we could change camcorders.

For wired remotes we again look to the manuals... years ago, LANC wired remotes were common in the mid-high end consumer cams. Time passed. LANC at the consumer level went away. A few had a combo AV/ wired remote and that seems to have fallen off, too. Some of the Sonys currently allow integration with a "wired remote" on a tripod, but I am not aware of any extension cables or ease of removing the controller from the tripod (if that is needed).

LANC is generally at the prosumer and pro levels. Camcorders that use LANC use the standard 2.5mm 3-conductor connector, extensions are readily available - or easy to make. These camcorders are well outside your budget.

I've not seen "pause while recording" on any wired remote.

If you tell us what the application is, there may be other ways to do what you want.