I’m having the same problem right now 
I set all the dependency chain to Development (or Published for Anvil Extras). This is fairly easy, especially today that the UI will tell you if and where there are dependency conflicts.
Then I set all the dependency chain back to Production (or last tested for Anvil Extra) before merging.
If you merge while some of the dependencies is not on its “standard” setting, you risk to break other apps. So I decided that “standard” is Published and I only set them to Development for the shortest time possible.
It used to be more difficult: About Development/Published dependency management
Now it is easier to get it right, because dependency conflicts are highlighted immediately after you change a version, but conflicts on apps you are not editing are not highlighted.
I think merging two branches should only be allowed if the result does not introduce errors: [Beta] Make sure apps don't break at time of merge because of different dependencies, or a big warning message should show the details of any changes.
… but I’m not sure it is even possible.
I asked for some controls on dependencies also here: Dependencies on Branches and Tags - #8 by stefano.menci