Advice on exchange-completion for a double-rent situation (a FTB buying from a BTL landlord)

Hi guys, I'm renting with 2 month's notice, currently trying to buy from a BTL landlord.

I initially (probably naively) assumed we'd exchange, then the seller would give his tenant 2 months notice, I'd give mine, and we complete after 2 months.

Reading older posts about long periods between exchange and completion, everyone agrees that's a terrible idea, and we shouldn't exchange until vacant possession.... But also not to give my notice until exchange.

That means I'd be paying for 2 flats (at London prices!) for 2 months, and that's a shed-tonne of money that could otherwise go to deposit....

Suggestions on how to manage this? Anyone else successfully managed that transition? Thx