I'll answer your specific questions first, and them some discussion afterwards.
How could the fact there are double cousins get in the way of what I
already have done?
The expected centimorgan match range for double cousins will be double the expected range for regular cousins.
You will have to consider all the children of both Bridges/Nisbit marriages as potential suspects, just giving you more possiblities to be eliminated
Is there a way to find out whether I am a double cousin to these
matches or a true cousin?
I think you will find that out the same time you find out who your grandfather was. So, no help in the short term.
If I was a double 3rd cousin instead of a true 2nd cousin, my dna
matches with these three would be weaker than what they are ( roughly
That is true. The upper end for 3C on the green chart is 109 cM. Double that is 218, still not close to 300. [edit: I recently got a double 3rd cousin match at 185 cM. The same person is also double 3rd cousin to my brother, and matches him with 226 cM. These are in the same range as 2nd cousins, and closer than many of our actual, tested, 2nd cousins. ]
Where to go from here?
It would be helpful to know how some of your matches in this family match each other. Ask your matches for access to their DNA results. In my search for my own grandfather, I suspected I was second cousin to some of my matches, but the cM counts seemed suspiciously low. But, when I compared, my brother and I matched them, on average, more closely than those documented 2nd cousins matched each other. That gave me confidence that I was on the right track.
You should make a McGuire Diagram to keep track of things.
You can use DNA to rule families out as well as in. For example a 300 cM match is not close enough to be your first cousin, therefore the matches' grandfather is ruled out as being your grandfather.
Don't fixate on your matches' close families too much. Your grandfather's immediate family hasn't taken a DNA test yet, or you would have a really close match, and you would be done.
You may come to a point, as I did, where you need to start buying DNA tests for people in your suspected grandfather's family tree. When contacting them, remind them that this is a saliva (not blood) test and work out a deal where you have full access to their results. You may have to register the kit for them, as they may be non-computer savvy and may have little interest in genealogy.
Test the oldest living generations first. If your applicable parent is alive, obviously, test them, or any living aunts or uncles. The same thing in the Bridges/Nisbit family. If any are alive in your parents or grandparents generation, test them first.