Transportation questions

You could consider renting the car in Bath and visit some villages on your travel to Stow on the Wold. Then, your travel day isn’t “wasted,” but you’ll be making use of your car the minute you pick it up. That leaves you two more days to make use of your car as you explore. You are right, using public transportation in the Cotswolds wouldn’t leave you with many options. We very much enjoyed hopping from village to village with our rental. On your way to Liverpool, you could visit Stratford upon Avon and drop the car there (or in Birmingham) for an easy connection. As you consider your budget, think of this transportation as a small percentage of your overall trip cost and increasing the value of your limited time there.

I don't think $300 is too awful to give two people a chance to roam around a tough-to-access area like the Cotswolds. I don't know whether you've considered this alternative:

Stay in Moreton-in-Marsh, which is sort of a bus hub for the area. Take a one-day small-group (van) tour of the Cotswolds. I was happy with the GoCotswolds tour I took in 2017. The current cost is 50 GBP per person--which seems like a good price. You get to see a bunch of small villages.

Then, on your second day in the area, see a bit more by taking advantage of the public bus service available from Moreton-in-Marsh. Maybe use a bus outbound and one or two pre-booked taxis during the day so you can see more than one destination and get back to M-i-M without worrying about bus schedules. Going the bus + taxi route would also be good if you want to do some village-to-village walking.

Since you want to spend three nights in the area, a car will be a lot easier to work with.

Hiring a car at Bath and returning it at Birmingham International or Stratford-upon-Avon would solve a lot of problems

I checked Enterprise for pick up in Bath and return to Birmingham. The most economical choice is a small Renault van - 100 less than a car. Train transport from Birmingham to Liverpool is also the most economical, rather than from a smaller town.
From what I've read, I need to purchase a railcard for 2 ride together or seniors because we'll also take a train from Liverpool to London. I agree that it's money we'll spent to travel around to see different parts of England! We're planning to take a coach from Heathrow to Bath upon arrival because it's seems like the simplest, although not the fastest. We will have been traveling for 25 hours by the time we land, so simple and easy holds great appeal!

Be aware that a van in British English is a commercial vehicle. https://business.renault.co.uk/van-range.html

What about hiring a car from Bristol airport then returning to Manchester airport? Might have more choices and more competitive prices.

yes, I agree. Be very careful by what you expect with "van". No windows and a cavernous space behind the wall after the 2 seats probably. Much better choices just up the road in Bristol probably.

https://www.bristolairport.co.uk/to-and-from-the-airport/car-hire

I just checked driving time from Stow on the Wold to Manchester. I didn't realize how short the drive is!! My husband is leery of city driving in England and I can't drive a manual. I have reassured him that driving in the Cotswolds will be rural and not congested city traffic so he's ok with that. Is the traffic light and easy to navigate from Stow to Liverpool? Are the routes straightforward or is it easy to miss a turn and get lost? Haha, we're seniors and like to blame poor navigation on our age!
I also checked car prices from Bristol to Manchester and Bristol to Liverpool - very reasonable!!! Thank you so much for the suggestion! Transportation from Bath to Bristol looks easy peasy as well! If the drive from Stow to Liverpool is easy enough for my husband, this is definitely the best plan so far!

You need to check the train fares before getting any Rail Card. A Two Together Rail costs £30 where you would need two Senior Railcards at £30.00 each. There are more restrictions with Two Together card as cannot be used before 9.30 weekdays
You need to spend £90.00 to break even per card
If you buy Advance Single tickets this is unlikely if you are only to make two journeys. With these tickets you have to travel on booked trains
Suggest you consult www.seat61.com for information on rail travel in UK including how to book tickets from abroad
The drive to Manchester is not recommended as involves some of the busiest motorways in Europe. M5 and M6 where congestion and delays commonplace .

Thank you for the information Roger! Maybe picking up car in Bristol and dropping off in Birmingham, then train to Liverpool is better for us. Car prices are even better dropping off in Birmingham.
Regarding railcard, can they be used for HEX? This plan would be 3 train rides: Birmingham to Liverpool, Liverpool to London, Paddington to Heathrow. Also we're planning to visit the Harry Potter studio as a day trip while in London,as well as Windsor. Could we use a railcard for those trips? We are not especially early risers, so we would only travel after 9:30 if at all possible.

Stay clear of Swindon.

Magic Roundabout in Swindon (the video of nightmares)

Wow!! Good to know! After looking at the map, A429 looks promising, through Cirencester.

I suggest you click this link for a Google map of the Bath area and then zoom down & look on Streetview to see what the roads look like - be it motorways (interstates in American), ordinary ‘A’ roads plus the country lanes. Also check out busy junctions and city driving. Please note that traffic is enabled so jams will show up live UK time. https://www.google.com/maps/@51.5509108,-2.3602536,49289m/data=!3m1!1e3!5m1!1e1

For longer train journeys such as Liverpool to London (Euston) - you get cheaper prices by Advance booking. For short hops - such as London (Paddington) to Windsor (change at Slough) - pay on the day for a return.

We will be staying 3 days in the Cotstwolds mid-April. We are stying at a B&B in Moreton-in-Marsh (taking train from London). We will be doing two days of tours with https://www.cotswoldsguidedtours.co.uk/. You might look into doing a day or two with them rather than renting a car. I have been very impressed with their response to questions and flexibility.

I did spend a few hours in Bath a few years back. The Baths and the Royal Crescent are worth a visit but otherwise it seemed a very expensive town for food. Nearby Bristol is much more interesting.
Its not a place i would rush back to

A429 looks promising, through Cirencester.

That's not just any "A" road. That's the Fosse Way. Of Roman days. That's why it is generally so straight.

That's amazing!! Thank you for the info.