My DD is 13. We have fully and amicably shared custody, no maintenance paid by either side for this reason, thought for school and work reasons she currently spends slightly more time at her dad's.
Her clothes, like all of her possessions, are seen to belong to her, not the house, and she freely traanports things between her two homes as and when she sees fit.
Her dad has a lot more money than me, and therefore buys the majority of her clothing I would say, though my family and I often buy her things too, as and when.
Her dad finds it challenging to buy her stuff like bras and pants, understandably, so that side falls to me generally, which is fine.
However I have recently become slightly frustrated, as their house seems like a black hole in which underwear goes to die! My mom and I spend a small fortune on knickers and bras for DD (she has some slight sensory issues and so is picky about the type of fabrics she will wear regarding these items - fair enough!). I spoke to her about it this weekend and she said she has asked her dad about this, as she too feels they disappear when put in the laundry. I wonder if he is using the wrong cycles, and wrecking and binning them....
The solutions my mom and I have come up with are:
A) to give DD more responsibility for her own laundry in both homes (she is fully capable of this).
B) that I give her an annual clothing allowance, to incentivise her to prioritise care for these items, to a certain extent. Plus it's always good to encourage independence and help them learn budgeting skills.
(Obviously if she actually runs out of both knickers and money, I will step in - but I would like her to take more responsibility if possible).
Sorry, that became a big ramble about my own situation!
If I were you OP, I would recommend to your partner that the kids take responsibility for their own clothing and are allowed to move it between houses, and that you give them a clothing allowance of whatever seems sensible. If they are anything like my daughter, they will give going shopping relatively often with friends, and are old enough to know what they need clothing-wisw, and when. I am going to give DD's in the form of Primark vouchers, as that seems to be her clothes shop of choice!