Things have changed a teeny bit recently. Before COVID, pets were an absolute no no. These days, LLs are not meant to refuse pets (it's guidance only though).
I emailed a few agencies recently and said 'we've got a cat, she's a good girl who never tears up carpets, but we did previously have a cat who did and we just replaced the carpet and the landlord never knew about it.' I had also put on the email that we have healthy savings.
The agent called me and I reminded her about the cat, and she said 'it's fine, we judge these things on personal circumstances.'
I took that to mean 'we know you have money and we trust you will just replace any damage.'
I think someone with an entire house sale in the bank is always going to trump a renter with little savings, and you should make that known on any inquiries.
I have also seen several ads that say pets will be considered at an extra cost of 25-45 pm, which I reckon is probably covering the extra cost of landlord's damage insurance including pets.
So, I definitely think it's worth making some calls on to test the waters.
Bizarrely, I came across a few last year who told me 'dogs fine, cats not.' Yeah, I have seen dogs eat entire walls, but okay...!
I think it could be a smart move to rent for a few years and wait for lower prices.
NB, I am not in London, most of my inquiries have been deep countryside where pets are more expected, but I still think your financial situation should put you somewhat ahead of other renters.