I have an A7SIII and just tested this on FCP 10.6.5 and Ventura 13.2.1 on an M1 Ultra Mac Studio. The FCP stabilization seems to work fine in both proxy mode and original media mode.
Keep in mind there are separate render files for proxy vs original media mode. If you want the best indication within FCP about the stabilization effect, do a CTRL+R to render the timeline to cache (this assumes background rendering is off). That must be done separately in proxy mode vs original media mode.
If exporting, that should not be required to obtain stabilized material. However there are two phases to stabilization, the analysis phase and render phase. In some cases if the analysis phase gets stuck it might export that without stabilization. If you render the clip or timeline to cache with CTRL+R and carefully observe if the "render dots" all vanish above the timeline, it should be OK.
You do not have to separately stabilize the clips in proxy vs original media mode. Applying stabilization in one mode should apply it in the other mode. However when first applying stabilization it's good to observe the viewer and wait until the message "Analyzing for dominant motion" goes away before rendering that to cache with CTRL+R.
Note if you are concerned about stabilization, the A7SIII records gyro stabilization data which can in some cases be used by Sony's free Catalyst Browse utility to stabilize the material. This normally requires the A7SIII IBIS and lens optical stabilization be off. However at 60 fps and 1/120th shutter speed it is very effective. There is a new FCP plugin available on the Apple App Store called GyroFlow Toolbox which can use Sony gyro stabilization in FCP. I have not yet tried this myself.
Sony gyro stabilization typically works much better at 60 fps and 1/120th shutter speed, but you are already shooting at that. I suggest you also evaluate Sony's gyro stabilization. You can combine that with FCP's NLE stabilization.
