do I want to pack a bathing suit if I am going to the Lake District

It's perfectly possible to swim in the lakes, but even at this time of year it won't be warm. My wife would be straight in but she loves open water swimming in cool water. If your idea of a swim is to dip in lovely warm water then leave your costume at home.

Lauri,
For the Lake District-
If you are staying in Keswick there is an indoor pool and leisure centre at Keswick (past the Museum, by the old Railway station).
If the weather warms up a bit again lots of people bathe in the lakes, or in the tarns when out walking the high fells.
In Summer it's far from unknown for people to go skinny dipping in the remote high country tarns.
Grange in Borrowdale, by the stepping stones in the river, is a perennial favourite for bathing.
Although probably out of the area you are staying in, around and to the east of Penrith, there are a number of community run outdoor heated pools- an interesting very local tradition. The closest to Keswick is at Greystoke. If you have a car it's 20 minutes drive. Greystoke also has a lot of history to it, especially the Church. None of this is in any guidebook of course, Rick or otherwise.
You can't get to Greystoke by public transport.
I'm sure any of those pools would love to welcome visitors.
The easiest to reach on the train is at Lazonby on the Settle Carlisle line, a short walk from the station.

Trying to pack light, but I do love to swim and don't mind cool water so better to have the option than not to have it. Thank you.

Well a swimsuit doesn’t weigh much and being from New England you will be used to cold water so bring it if you wish. I have found that I have rarely used my swimsuit when I travel except for southern France or the Adriatic.

It doesn't take up much space surely so just take it if there is a genuine prospect of you swimming