You can find out if there is a mortgage on the property by paying £4 to the landregistry website. It will show who has a charge over the property.
You can then attempt to find out from the lender (no idea if they will tell you) if they have given permission to let.
If they have, they are bound by your tenacy agreement
If they haven't, they're not.
You are unlikely to get the agents to change the tenancy agreements - thay like to have them all the same so that don't have to check for changes every time they do something.
If you had the fund available, how about this as a counter suggestion to their 6 months requirement...
You will pay 3 months up front and a deposit equal to 4 months rent. That way, your deposit will have protection from one of 3 government approved schemes (preferably DPS), if the property is repossessed it will be at least 3 months before you're evicted but the down side is that you will have to pay rent from the 4th month. Obviously, at the end of the tenancy, you get the deposit back.
If you like the information/help provided - Gizza Job
(People under my age won't understand the significance of those last 2 words)