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 ...
OK, so am I understanding it correctly that you have the following set of relationships:
?1 --- A --- B
C --- D E --- ?2
where F is you, and G is your cousin?
The easiest explanation for a match of 1760 cM between F and G is if ?1 = B, that is, your dad (B) is actually your aunt's (E's) full sibling, ...