Tommy Edman
Honestly, the odds that they would move Tommy Edman are the lowest of anyone on this list (at least at the trade deadline), but I still think it's worth thinking about if they want to get creative.
If the Cardinals did want to go out and get a top starter or an impact right-handed bat, Edman could be a piece that the team they are looking to deal with would be interested in, especially if that club is a contender or wants to contend in 2025. Remember the Harrison Bader for Jordan Montgomery trade in 2022? Bader was injured at the time of the deal and was significantly less valuable than Edman, so I could see someone wanting to target him this month.
The Cardinals have always been bullish on how valuable Edman is to them, and his absence has been felt in a major way this year. They may have a hard time convincing themselves that it is worth losing him, but I'll make the case for a scenario where it could be.
Michael Siani and Dylan Carlson are capable of holding down center field for the remainder of this year. The Cardinals already plan on having Siani continue to start against RHP when Edman returns, so Carlson would just play in the 30% or so of games where a left-hander is on the mound. If Nolan Gorman turns things around offensively soon as well, then that natural spot for Edman does not exist.
While Carlson is the player you'd prefer to move, they would get a lot more value from moving Edman. If Edman could get them their next Jordan Montgomery, is it more valuable to make that move and keep Carlson rather than dealing Carlson for whatever they can get and keeping Edman?
Edman is also only under team control for one more season. so his value will only decrease as time goes on. He clearly helps this team now and in 2025, but there could be trades that help upgrade the Cardinals further in a way that keeping Edman around does not make up for. I also still doubt that the Cardinals will need to keep Edman around beyond 2025 either way.
I don't see this as a likely outcome of this deadline, but hey, we all seemed to be surprised by the Bader deal when it happened, and he was similarly displaced by Lars Nootbaar and Dylan Carlson right before that deadline as well. Maybe the performances of Masyn Winn, Alec Burleson, Brendan Donovan, and Michael Siani have done the same for Edman's role on the team.