How do I ask him if he wants to be FWBs with me without straining our friendship or creating an awkward situation between us if he refuses to?
I'm going to go out on a limb here and say that there isn't really a risk free way to do this and suprisingly enough, gender and sexuality don't make as much of a difference as you might imagine.
Anytime you proposition anyone, you're taking the risk that you may be rejected. If you proposition a friend, you're adding the risk of straining and/or making the friendship awkward. These are just realities of any sexual proposal.
You can test the waters with casual flirtation before making a proposition, and that may give you a better picture of how the person will react, but it definitely doesn't tell you for sure.
Friends with benefits situations are inherently risky no matter the gender or sexuality of the folks involved. Someone may develop deeper feelings, someone may be cheating on a partner (nudge), someone may want to end the arrangement while the other doesn't, and the list goes on and on... No strings attached sex between friends is rare, there are nearly always strings attached, it's more a matter of whether the people involved are aware and emotionally mature enough to handle that.
Now, on to the other part of the the question, the same-sex-fwb part. I've read enough Dan Savage to know that there are plenty of straight identified men who enjoy bro-jobs, buddy-baiting, and other same-sex-sex while still maintaining their straight identities. Seriously, straight people write him an awful lot with these kinds of questions... An awful lot...
Dan Savage mentions the idea of straight male fragility, in the piece I linked above.
Male heterosexuality, in this way, is a lot more fragile than female heterosexuality or male/female homosexuality. But with that said... a lot of the white guys (and a lot of the not-white guys) out there ... are closeted gay or possibly/probably closeted bi men. I kept waiting for the word "bisexual" to pop up in the press release for Not Gay but I didn't see it in there. The men Jane Ward studied might not be gay—gayness could be ruled out in some cases—but straight-identified, married-to-women guys who have sex with other men are likelier to be bisexual, closeted or not, than they are to be straight, fluidity or otherwise.
And honestly I tend to agree... Many straight men have some serious hangups about being perceived as anything other than straight men, even when they actively seek out not-exactly-heterosexual sex. The stigma, and fragility of male heterosexuality makes it really hard for some guys to experiment, without the fear of being forever labeled, or to embrace the idea that it's ok to be something other than a straight male. As Dan Savage put it:
While I believe a guy can have a same-sex experience without having to identify as gay or bi—straight men should have the same latitude on this score that straight women enjoy—straightness is so valued (and apparently so vulnerable) that some people can look at guys who put dicks in their mouths at regular intervals and construct book-length rationalizations that allow these guys to avoid identifying or being labeled as bi, gay, or queer.
To be clear, I think the fragility of male heterosexuality has more to do with the way society tends to label people, than it has to do with the way these men label themselves. To use myself as an example; I am a cisgender man and I identify as queer, but I also date women. Society doesn't label me as being any less queer for dating or sleeping with women. On the other hand, before I came out, the moment I began to even consider dating people who weren't cisgender women society was very quick to label me as gay, bi, or queer.
Notice how once queer I'm considered queer regardless of who I'm dating or sleeping with, but as a straight/heterosexual man even having thoughts or fantasies will change the way society wants to label me? I don't think that's right, or fair, but unfortunately that's the way things are at the moment. That's the fragility of male heterosexuality.
Now with all of that in mind...
I think this is probably a bad idea... Unless you're willing to risk the friendship, each of you are willing to be honest with your girlfriends, and you're both willing to be honest with yourselves about what you're doing, you probably shouldn't.
If you really feel the desire to experience this kind of thing, talk it over with your girlfriend, and then maybe find someone who's not a straight, close, friend.