This is a trap for young players.
I will never ever consider to use a "free" pcb program when it only serves to lock me into some vendor's manufacturing capability. This used to be a common practice not too long ago. I vaguely remember some PCB design suite that was "free" to use, as long as you used "their" manufacturing capability, but it at least had an option to buy the Gerber files from them if you wanted them.
10 years forward and KiCad has evolved to such a state that it's more then capable of any but the most complicated boards, and its completely Open Source and therefore free to use, or donate as you see fit compared to how useful it is to your company and if you have budget left. More money does help improvement of KiCad!
That said.
If some company refuses to give you the Gerbers of your own design I would completely ditch that company for any future projects, just out of principle. I refuse to be held hostage, and redesigning a board (layout) is a small price to pay for my own freedom.
But then again, I never would have fallen into that trap in the first place.
How much effort would it be to re-design your board? Having a bare PCB as a guidance for component placement and track routing can be a big step to reduce the reverse engineering effort, especially with 2 layer designs. If you have 4 or more layers it becomes more difficult.
Here are some pointers for reverse engineering a PCB.
https://hackaday.com/?s=pcb+reverse+engineerSome of the techniques used are flatbed scanners to scan pcb tracks with minimal distortion.
Sanding off PCB's to expose multi layer copper for scanning all layers.
Making high resolution photographs and layers with Gimp (photoshop, etc) and use different layers and transparency to check connections.