Our client is reluctant to allow permissions ...
As it their right.
You have no mandate to be dictating the security settings that your client "has" to use.
... this has fixed the problem on our test environment
This has only masked the problem because the problem still exists at your client's site and, let's face it, that's the only place that matters.
You must have a representative testing environment, i.e. the same as your client[s] site[s], or your tests are practically worthless.
They persist that this was not necessary before deployment of our supposed fix ...
"Supposed fix". Interesting turn of phrase.
Or, perhaps, a realisation that your change wasn't the right thing to do.
... and that it should not be necessary for them to change their security settings ...
And they are absolutely right.
You've made a code change that rectifies the reported defect (about table selection) but have introduced a new defect whereby your code does not conform to the client's chosen security settings.
The fix really have nothing to do with permissions
Possibly not, but it's introduced a new defect that clearly does.
we would like to better understand the problem, before making a decision of what to do next.
You don't have much of a decision to make.
Find another fix for the table selection problem that the client is happy to accept.
I am thinking that maybe it has something to do with build parameters, such as compiler versions used.
Or that it might have something to do with environmental settings (64 bit vs 32 bit).
No; I would look at that ActiveX Control first.
Did you introduce this control as part of the table selection "fix"? If so, that's almost certainly your problem.
If it was there before the "fix", then you have other problems.
I have no idea of where to start, or what to compare.
You don't use a Source Control tool to record all changes made to your source code? Project Files?
Is there a tool that can be used to compare compiler settings used to create a dll? (the changes was in a dll).
Practically every Source Control product on the planet comes with some sort of version comparison tool.
If you're not using them, then you really are flying by the seat of your pants.