Resident alien...release equity to buy rental -tips?

[Cat Bulmer]

Hi all, I'm a British citizen, currently living in AZ (been here nearly 2 years). We sold our London flat and bought a house last year. Now we would like to release some of that equity to buy a rental property - but I don't have the credit score to take two dings, i.e. one for the HELOC and one for the rental mortgage.

Property info: 

#1 main residence, current value $1m, outstanding mortgage $400k

#2 current value $600k

So, ideally I'd like to have one mortgage for $1m, to cover both properties - as that would mean only one mortgage, only one enquiry = increased probability for approval. 

Any suggestions on what I could do?