Your bicycle frame has a traditional BSA bottom bracket shell based on the link you provided about your current BB, which means you'd want to switch it for a BB-BSA24R-style BB. These are widely available for little money; Shimano for example has a budget option available as BB-RS501.
Random link:
https://www.aliexpress.com/item/2251832565113180.html (select Color: RS501 BSA)
The store that sells the PR3 crank has an even cheaper version (the above link has extra shipping):
https://www.aliexpress.com/item/3256805811238798.html (Color: B-BSA-24(68 73MM))
You'll also need tools to remove the old BB and install the new one. I don't know what tool is needed for the old one but it's likely an ISIS 20 tooth spline socket, readily available on AliExpress. For the new one you'll need 16-notch 44mm bottom bracket tool, also available all over the place (typically in either a cup or a wrench configuration).
And for the crankset you need a crank arm cap tool, but that's likely included with the crankset itself.
Regarding the question of just keeping the square spindle, that certainly seems like an option as well, especially if the current BB has plenty of life left on it. One question is how well the chainline would line up, I don't have any experience there.
One general concern is whether your frame can accommodate wider tires - what tire size do you think you'll be able to fit?
