Does the difference between a marketable limit order's price and the bid-ask price influence the priority of the order?

"At exactly the same time"

Your question is meaningless.

In the database, there's just a sequence for orders. (1, 2, 3 etc.)^† They cannot be at the "same time", time is not used and the concept is meaningless. The sequence numbers are unique and ascending.

So as a general concept there's (a) price ordering {no connection at all to time} and (b) sequence ordering {no connection at all to time}. (And indeed (a) comes first then (b) second in the spirit of your question.)

Setting that issue aside, the answer to the spirit of your question is "No": the limit figure is not used in any way, whatsoever, to decide between two possibles, only the sequence mentioned in the 2e para.

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^† {Time, per se, is irrelevant; it would be commonplace and ubiquitous that many orders have the same time-stamp, whatever resolution is used on whatever platform; it means nothing: the sequential ordering is used to differentiate.}