If you haven't moved over, what's stopping you from making Monzo your main bank?

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I believe Monzo can help you with your first two points;

You can deposit cheques into your Monzo account, you just have to post at Freepost Monzo

Monzo has increased limit for some so you have to ask and they might do the same for you;

Current account limits insufficient for extraordinary purchases

I can confirm that I did this and that the car is fast The process is essentially reaching out to customer support and letting them know you need an increase. We’ll arrange for a specialist to get in touch and ask a few questions!