Can I purchase any wheel with the right size & bolt pattern?

Yes, in general, same diameter, width, bolt pattern and offset, most of which are standard for trailer wheels, will work.
You will need to wade through the multitude of responses that are overthinking it. As many of the folks here only have limited, recreational experience with trailers and do not fully understand, and therefore are basing much of their knowledge off of very limited experience.

Subjectively, unless the second spare is just so you can sleep better at night, fix the issue that is making a second spare tire seem like a necessity.

Your title says you are looking for a wheel. Your post says tire. If wheel/rim look at the inside of your current wheels. You should see something stamped in it stating max inflation value. Make sure that matches on the new one also.

It should be fine, especially as a second spare.

Shopping on price alone is not wise, there are a lot of unsafe cheap tires out there. Go with a quality brand from a reputable tire dealer, the rv and auto dealers are generally the highest.

You need to known the offset ( if any) and the bolt pattern.. Mine is 8x6.5