Are you a bisexual guy in a relationship with a woman? Because this is a HELL of an assumption.

Compared to the spectrum of relationships men could have, I'd say a heterosexual relationship comes with the most "privilege": there is no need to make grand announcements to family members about your sexuality, you can freely express your love to your partner and no one finds it 'icky', you can bring your GF to thanksgiving and not have to deal with a crazy uncle making homophobic remarks, no family members that stop talking to you, etc. etc. Are we in agreement here? Assuming we are, what's the next best thing? The next best thing, IMO, is a bisexual man being in a heterosexual relationship with a woman. Assuming your SO is totally fine with it, it brings no major hardships.

and it can be aggravated by being in a het-appearing relationship

I'm talking about the general case were you're with Sally because you want to, not because your mom wouldn't approve your relationship with Jake. A bi man forced to be with a woman because of the circumstances would have many, many (understandable) issues that no one would want to deal with, OP however clearly indicated that this wasn't the case he/she was talking about:

"I was referring to men who are in the closet about their bisexuality, and are presenting as totally straight because they're in a relationship with a woman."

having to watch what you say

I like pegging. If the conversation of pegging comes up in front of my mom, I'm not going to offer an opinion on it. My wife also will probably remain silent. This is not a terrible burden to carry, in fact, most people keep their kinks private.

where your eyes linger,

You're telling me you cannot control yourself and stop checking out cousin Tommy's butt in front of grandma?

can be a strain on anyone

Sure this is a problem but no where near as severe as being in the closet about being homosexual, where you cannot express your love openly "no, I'm not bringing anyone to thanksgiving dinner", where you have to lie about your SO "oh, he's my roommate", hiding, etc.

there are legitimate reasons why they'd be wary

Meh. I'm not convinced it isn't simple homophobia and/or expectations that their mate will behave in certain traditional ways (i.e. similar to way her father or father figure behaved). "Gay people are OK, but I'm not going to date a guy who likes sucking dick no matter how much he likes eating pussy"