This issue is being substantially overthought.
The cost of renting a car from Heathrow to Bath merely to park it up there for several days will substantially outweigh the cost of the train tickets, let alone the bus tickets which many forum travellers use to get from Heathrow to Bath.
Then we have the issue of driving on a busy freeway/motorway straight after arrival on a long haul flight
There is an hourly direct bus from Bath to Bristol Airport so that is a non-issue.
If Avis are 2/3rd the cost of Enterprise, then why not use them? There is a bus several times an hour from Leeds/Bradford airport to Leeds City for trains with multiple departures an hour to York.
A very easy journey.
The bus is £2 each and a train ticket is often less than £5 each.
There are also two fast bus journeys an hour from Leeds to York.
With Avis I can see mid size automatics for around £440 a week.
With Enterprise the office is at York Railway station. In May, for a week Bath to York I can see their prices from £326 for an Economy automatic to £410 for a Standard/Standard SUV and £630 for a people carrier.
So there is little difference between Avis and Enterprise (Enterprise are probably a bit cheaper overall), and where your $1000 a week comes from is a mystery.
There is also a lot of over hype on this forum about needing a car in the Lake District. For a standard tourist itinerary you don't. In many ways it is a liability and a hassle. Frankly both the local authorities and many of us locals are weary of those who insist on driving everywhere in the area.