If we believe Amaimon is spying Shiemi, and it is an order from Mephisto, then Mephisto already shown special interest in her. It is convenient, but the whole Shiemi's family secret just shown in the story after Chapter 84, so Kato can't mention it too early.
And I don't feel like Shima's conversation being just casual in Chapter 88. It's suspicious because he seems intentionally trying to end that topic
Didn't Amaimon make it clear that it was rather personal by mentioning the old bride thing, making it not ambiguous? I don't think there's much to speculate about aside from the reason why Mephisto would let him free in the school
There are actually two questions:
1.Why did Mephisto release Amaimon? Or more precisely, what does Mephisto mean when he said "It's about time." in Chapter 82?
2.Why did Mephisto ask Amaimon to be a (good) student
in Shiemi's class? I think Amaimon made it pretty clear that he doesn't want to be a student, but he promised Mephisto.
Maybe there are many possibilities for Q1, but I think the most reasonable answer for Q2 is Mephisto asked Amaimon to spy Shiemi.
About the bride thing, I thought it's quite evident that he said it in order to have a fight with Rin.
I'm saying Renzous typical normal dialogue is horribly similar to his spy-influenced speech, therefore it is fair to argue either points.
But I don't think it was really normal. What happened in Chapter 88 is, he has completely opposite reaction (compare to others) about Shiemi's quit. If that's not suspicious enough, later when Izumo brought the topic again, he rapidly changed his opinion to be rather vague.
there's more than one thing suggesting that Shima knows little to nothing.
A little thing that you may want to consider as well is that Shima prepared his Krik to possibly engage with Amaimon after Shiemi pushed him, which is something he would not do if he knew much of the situation beforehand.
For me this is actually another evidence that maybe he knew something : his attitude about Amaimon was a 180 turn between Chapter 83 and 84.
In Chapter 83 he was like "WTF Sir Pheles is thinking?", while in Chapter 84 he was like "Maybe strong girls are his type

" , and effectively switch their topic. In my little theory that's because he learned Amaimon's mission from Mephisto.
I'm going to drop my 2 cents that Renzou for once isn't lying and actually doesn't know Shiemi's secret, as I think that Mephisto sent Amaimon instead to observe Shiemi.
Yes, but in my theory he wasn't lying, technically