I don't understand your post.
In Point (1) you seem to say the previous occupant used Hi-Velocity and that you also use Hi-Velocity at your previous address.
Did you cancel your contract with them there, so now have no actual relationship with them?
If yes, then surely all you have to do is sign up again if you want to. There would be no tag issue for them to sort out. Though I expect you will have paid a Cease charge at the old place and get a connection charge at this one. Most ISPs do free house moves. Too late for that now.
If what I ask above is correct, Point (2) doesn't really make sense.
"Jump while I can and go with another ISP (assuming I can do this). I am assuming that if I do so I will have to pay HV for a month or so but don't have the trouble of getting a MAC code from them".
You haven't got any jumping to do. But you are right, you don't need a MAC.
Why should you have to pay Hi-Velocity anything for a Tag they have left there after the previous house-occupant who had a contract with them has gone?
You just ask them to remove it. They probably won't

. Who told you it is still tagged? Is that who you would prefer to sign up to?
In any case if you sign up to any other ISP of your choice then the new ISP has a BT Wholesale phone number to ring to get the tag cleared. Though most ISPs seem ignorant of this.
Just today, someone who could not sign up to anyone for this reason, (he cancelled his Sky contract a year ago and they left the line tagged), got help from a totally different ISP on these forums. The tag was cleared by them in under 75 minutes, with no obligation.
Re heavy use, what do you call heavy? And what times of day does the heavy use happen? Is it scheduled while you are asleep?
There are a number of good ISPs out there but that's the sort of info we need to recommend which. Also how much are you willing to pay, on top of the line rental?