Divide the budget by the number of staff, give them it as cash in an envelope.
Knowing what the op means about miniscule budget, in my school that would mean about 28p per person!
My school set up a wellbeing team in September after feedback from staff during lockdown. We did a few bits last term - mostly setting in stone expectations about trying to get the work/life balance lance right, and escalating up (as far as the principal if necessary) if folk were struggling with that. We have bake-offs that contribute house points (and we get to eat 'clean' home baking if you know you know), mini sport events after school that the kids can hang around to watch. We have a healthy living ambassador that ran a kind of 'get out there' type thing - send a pic of your daily exercise which was uploaded to the school twitter feed, which actually encouraged lots of kids to get some fresh air.
Plans are now in place for virtual bake-offs as we're now on a rota, and it's not just cake this time. Someone is setting up a weekly new recipe challenge where we each make a meal for ourselves/our families that will broaden our regular meal choices. There's going to be a quiz (yes in the evenings, but really what else is there to do, and it's sooooo good to see familiar faces that you might not see for weeks on a rota) for those who are into quizzes.
NICE TEA AND BISCUITS IN THE STAFFROOM is a must, and relatively cheap
Our school has access to a free counselling service for all staff, would be worth promoting that if you have the same. I know a few colleagues used that last year and are feeling stronger going into this period.
Open feedback to slt is useful - we have a postbox type thing which has seen lots of subtle changes in the last few months - slt calling into offices and classrooms for a genuine 'How's it going' chat rather than just power walks and lesson observations.