Of course no one can stop the OP from going ahead and getting the PW symbol for a tattoo. The OP may also find some Poles and numerous non-Poles who don't see any problem with it being used by anybody for whatever reason. Sure they may recognize the PW and know a little bit about its history but they have no personal connection to it so it doesn't matter to them who uses it.
However, Polish veterans are very sensitive about it being inappropriately used by other persons and organizations.
Indeed, the Veterans of the Warsaw Rising have sought to patent the symbol:
Veteran Andrzej Gladkowski, vice chairman of the Association of Warsaw Insurgents, told Polish Radio that the emblem is being used for inappropriate purposes.
The reason cited by the OP for getting this particular symbol has nothing to do with honoring an ancestor who fought in the resistance movement. It also has nothing to do with any direct involvement or support for Polish wartime veterans.
Rather the motive is just a personal one which is completely unrelated to the service and sacrifice of those Polish veterans who fought to liberate Poland from the Nazis.
The OP stated:
The main reason I want the Polska Walcząca tattoo is because I recently recovered for depression. And I want a reminder that I have fought it before and I survived despite many suicide attempts.
It's fair to say that tattoos are used as conversation pieces. Obviously the OP wants the tattoo to draw attention to himself and in turn raise awareness regarding his serious medical illness which he feels he has currently recovered from. However, there are more appropriate ways to go about this.
For example, the OP can get a tattoo of an "awareness ribbon" which is used by support groups for such issues like depression and suicide prevention.
Here is a list:
Better still the OP can volunteer his time and/or donate the money he would spend on a tattoo and give it to a local charity which helps people like him who also suffer from such problems.