I think the pill can add to the complication of DVT which is why (I think) they ask you to stop taking it! But I’m sure if you said ‘I understand the risks and I would rather stay on the pill’ that they would let you.
They just don’t want to be liable that’s all.
But on another note, perhaps now is a good time to look into other forms of contraception. I have a coil after years and years of being on the pill, and I know it’s not for everyone but I honestly wish I’d switched earlier, so much less hassle ! Just a thought!