How to do a budget holiday on Mykonos, Greece

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We are going for a week in July, traveling with our friends who are gay. DP and I are very open and are excited for the holiday!

We got really good flights there and back (£220 with luggage) and found a cute old fashioned taverna in the town that has rooms a few doors down. £100 a night for a small apartment.

So far we have booked the cheapest possible. But I didn’t realise QUITE how expensive it was. Just looked at some menus online and a glass of wine €15 etc. One place someone recommended charges €100 for sunbeds for the day!!!

We are thinking of taking our own umbrella and towels with us and doing a cheapo and not paying for rented loungers.

We were thinking of taking some staple foods with us too and self cater.

Any tips on how to save money otherwise?! We do want to enjoy ourselves still

You lost me at 'who are gay'.

Oh dear! Your opening paragraph does read as if you’ve just dropped in from 1982. What a strange thing to say.
Anyway. Mykonos is very, very expensive, it’s also small with a small population so there’s no escaping the prices, If your friends are flusher than you, you scrimping your way through could be a bit of a bore, it’s tricky.
Stock up in bottled water for your room so you aren’t buying drinks for thirst so you can make them count. Take cereal bars in your luggage and stick to the cheapest main courses when out, dodging starters and pudding. Yes to cooking as much as possible, use the excuse of wanting to experiment with local produce and crack open wine and light candles to make it less grim.
However do everything you can to scrape together more cash in the run-up.
Do you and your friends normally hit bars and drink a fair bit at home? If so, there’s no way they’re not going to want to hang out in the gorgeous bars in Mykonos. Stick to beers in the bars yourselves.
I’m afraid I can’t think of many worse places to be on a budget, particularly with friends, as you don’t want to drag them down with you.