Let HR know, and take the holiday. It will definitely help your recovery and you should be doing everything you can to get better - a holiday is a good step towards being back in “normal life” . Do not overthink it or worry about it. And don’t flaunt it!
Being on vacation during MH leave is actually a LOT better than taking it when you return to work. You can then use your annual leave to take say two afternoons a week off, so there’s less impact on your team and you can ease yourself back into work.
I had a colleague off work for 7 months, she was managing the team parallel to mine - so I was dumped with covering her work (not really my employer’s fault at first as HR had no idea she would just keep extending the GP sick note so they didn’t plan cover for months). She left me with a right mess because she hadnt been doing her job properly so that caused stress as she was trying to cover up.
I inherited her big mess, a team in crisis and was still doing my own job so I was knocking out 70 hours a week to fix what she had wrecked.
I did agree to it because I liked her, felt sorry for her MH, wanted her to recover and come back and see it would all be okay. As the months wore on she started posting SM about her “battle” with work related stress (aka being really a bit shit at her job and not telling our manager because she was too scared of being told off).
And then she comes back to work on full pay but only works four hours a day to help her readjust, does nothing much differently to show that she will manage things better and doesn’t even call and say “thanks for your help covering” and THEN demands to take two weeks holiday at our busiest time of year from her annual leave so she doesn’t get stressed again (blackmail).
I promptly quit without a job to go to and got a better job. I was wild with annoyance at her, she was utterly selfish and it didn’t occur to her what I must have gone through, sorting out her shit.
I really do have sympathy for MH - I’ve had plenty of my own struggles and I’m menopausal - but because now people use it as an excuse for being ineffective at work, and then flaunt their “struggles” like a bad of pride it makes it harder for people like you and me - good people, who want to do the right thing.
Take the holiday now, get better, return to work and look for another job which you can handle without getting stressed. That’s my advice.