The rust plugin tries to use rustc installed from the package manager, instead of the one from rustup. I noticed this because on the system's one I didn't have rust-src installed and rust-analyzer was complaining about that.
I can confirm that the rustup version is the active one, by running rustc --print sysroot from the same shell that I launched the lapce.
In any other editor I don't face this issue using rust-analyzer (ie. kakoune/helix).
OS: Archlinux
Lapce version: 0.2.3
Lapce-rust version: 0.2.3