There's no right or wrong answer. Have a close look at the course content/modules for the different courses. You can reach out to the different universities, and see if you can find out more information & and talk to current students for their experience on the respective courss.
For Bath specifically, last time I checked the Design and Mechanical engineering courses share the same first 2 years, then they differ. So you were able to swap courses before your 3rd year. Would be good to double check this is still the case yourself.
Since you live in Bristol, have a think about if you would want to live in you home city potentially for another 3-5 years. If you do, you'd already know the area, know whats good, probably be more comfortable with an existing social group & and you could save a lot of money if you lived at home .
However, going to uni in a different city can be a great opportunity for new experiences and growth, independance and just exploring what else is out there, and it can be a fresh start meeting - meeting new people and not being tempted to just stick with your old social group.
You'll likely have a different experience depending what you choose - but again, neither is necessarily the right or wrong choice, it depends on whats right for you.
It's can be very difficult choosing what course and which uni. Wish you best of luck, and I'm sure you'll be happy with whichever you decide! : )