Its not impossible to get diagnostic tests done outside the scope of a GP. It's generally very expensive.
It is no issue at all to ask for and receive an echocardiogram referral since it has no risk, just ask for a referral.
The unwillingness to do anything involving radiation unless warranted seems reasonable. Just do the echocardiogram and go from there.
I got a CT angiogram from a cardiologist here who started her own business specifically to do them. (It’s just her, a nurse, and a CT machine, in a trailer in a parking lot!) Insurance doesn’t pay for imaging like this until things have already gone very wrong, and she feels it should be used much earlier for people who have risk factors.
For example: https://radiologyassist.com/MRI.html