Make sure that windows updates and videocard drivers are up-to-date.
Also might be a case that certain redistributables are not installed.
If it won't help, file a ticket to the support team.

As Microsoft's documenation states, InitializeCriticalSectionEx syscall is supported since NT 6.0. So it's specific for Vista and later kernels. Looks like XP is no longer supported with the version of the game you are trying to run. May be earlier one could still be compatible.
About the solution, I didn't tried FTL but solved my problem of running Into the Breach on XP with the unoffical One-Core-API wrapper. It has to be installed into the system globally and conflicts with East-Asian languages support, though. And there is no localized versions of libraries as well.
Important. I strongly recommend to backup your Windows installation before using this method. Since you may encounter unexepected incompatibilities leading to unbootable OS.

I used to dual boot Windows XP and Linux Mint and whatever did not play on one worked on the other or then my laptop was simply too weak. I can tell you that FTL: AE worked just fine on Linux Mint in 2018 at least.

Go ahead and try Schwertz's solution first if you want to avoid the hassle of dual booting and the work of setting it up.

FTL works just fine on WinXP SP3 - the problem lies with GOG's Galaxy.dll - my version of FTL (downloaded from HumbleStore without that DLL) works without any problems.

However the GOG version can be "fixed" - see How to make the GOG version of Into the Breach run on Windows XP and XP x64 and note that you will need to follow the instructions in section 2.

This workaround should apply in other cases where GOG's Galaxy.dll causes similar problems, so feel free to raise a support ticket with them as well.

Well, looks like we need a proper Galaxy.dll emulator as a universal solution. It's the third case for me were you have to modify executable files just to get rid of the unwanted or even malicious influence of Galaxy.dll. 'Thank you' GOG...

Agreed, and I've pointed this out to GOG support. It is particularly galling when this code is included in the offline installer of a single-player game that would see no real benefit from Galaxy.

But we do have multiple workarounds - and asking GOG for a refund and buying a copy from HumbleStore (they offer direct downloads of DRM-free games) instead seems the best way of thanking GOG for their (lack of) support.

Last version without Galaxy.dll from GOG is 1.6.12. I just tried it on XP and it's working out of the box. If you can find it, it's your best shot.