In Bezug auf diese Frage gibt es drei Probleme.

Is there any contextual, social or grammatical difference in the preposition used with Bezug? I’ve seen “in”, “mit”, and “bei” used.

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There is only a very slight difference between “mit Bezug” and “in Bezug”.
It is both formal and mostly used in business correspondence:

“Mit/In Bezug auf Ihre Anfrage kann ich Ihnen mitteilen …”
Regarding your request I may inform you …

In these cases “Bezug” comes from “sich auf etwas beziehen”.

“Bei Bezug” is something completely different. Now “Bezug” is related to “etwas beziehen” (obtain something).

“Bei Bezug von 100 Stück erhalten Kunden einen Rabatt von 10 %.”
When purchasing 100 p. customers receive a discount of 10 %.

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With this particular sentence, I feel you can’t substitute mit for in.

I think mit only works together with an action, and “there is” is not really an action, nor is there an actor/agent.
The action is done while referring to something. Mit connects two things, the action and the context in which it is done, while in just sets up the context or topic in which to understand the rest of the sentence.

So, “in Bezug auf”, in my mind, translates to “with respect to” or “regarding” or “as for”, while “mit Bezug auf” would be “referring to”.

The english translation of this sentence (as found on tatoeba) is:
With respect to this question, there are three problems.
This works, as “with respect to” basically means “as far as this question goes”, i.e. setting the topic.

It would sound weird in english too, I guess, if you’d use “referring” here:
Referring to this question, there are three problems.
Who is referring to the question? The problems? No. Probably the speaker? But the speaker is not mentioned as an agent in the rest of the sentence. So maybe this works in english if you fill in the gaps and understand the sentence as:
Referring to this question, (I tell you, that) there are three problems.
instead of
There are three problems that are referring to this question.

Maybe this trick could also make the german sentence work with mit? I don’t know. To me it’s still weird.