I've been in polyamorous relationships for some time now, so I speak from a bit of experience here.
The first question to ask yourself before you even decide if this is a path you can take: You mention that your girlfriend believes that sexual relationships and emotional relationships can be distinct. Are you confident that you're wired this way, too? The last thing you want is to be in an emotionally monogamous relationship, having developed and acted on feelings for two different people. Many people can operate this way, but many people can't.
Second, realize that if your potential friend-with-benefits (pFWB) has romantic feelings for you, this is a risky proposition even if none of the other risk factors that I describe are present. It's very difficult for most people to engage sexually with someone that they already have a romantic interest in without those romantic feelings developing further. Some people can do this, but it usually takes a lot more experience navigating these kinds of waters than what most people have.
Third, realize the precedent that this might set. If you don't want her potentially having sexual encounters with other guys, now or in the future, this might be a path that you want to avoid. Later on, if her sex drive spikes upward and you just can't physically keep up, is it possible that she would feel like she was treated unfairly if she wasn't given the same option you take now?
Fourth, before you talk to pFWB, talk to your girlfriend about your plans. Tell her who you plan to approach and make sure there's no unknown drama waiting to happen on the end of your existing relationship. This could help to identify early any lack of trust between your girlfriend and pFWB, and if you detect a negative emotional response from your girlfriend at the thought but it doesn't seem to have to do with the specific pFWB in question, she may not be as okay with the reality of you having other sexual partners as she thought she would be. Some people are also known to suggest things like this to their partners, knowing that it would tear them apart. And even if she, while more emotionally healthy, was happy to see this happen, in a depressive state, she might be more vulnerable to feelings of jealousy, worthlessness, or 'being replaced' that might crop up in here, so be very attentive for that sort of thing. If any of these many potential problems crop up, full stop - abort mission! And if things go well, it might be worth asking if she wants to talk to pFWB, or if she'd be willing to if pFWB needed that. This might be a good time to establish rules and other boundaries - for example, to answer the question of how much you need to communicate with her before engaging with another partner, if you need to ask permission or just give notification, if this will open her up to sleeping with other people as I mentioned before, et cetera.
If you're sure that you can move forward so far, the name of the game is communication! The sexual encounter that you're hoping for might be no-strings-attached, but the conversation leading up to it has to be anything but quick and casual in order to protect the feelings of everyone involved.
You should begin this conversation with pFWB first by establishing that her friendship and her feelings are important to you. You do not want to come off as someone who just wants to use someone and toss them aside. Depending on pFWB's background, she might find the very suggestion offensive and demeaning no matter how you phrase it, so you want a cushion already in place to soften the impact in case that's how she winds up taking it. But don't dwell on this part of the conversation too long or she'll think you're working her up to something different.
You correctly identify that you want to make it easy for her to say "no." If she feels like you're trying to pressure her into something, or like you're holding your friendship hostage, or even like this interaction is in the moment more important to you than your friendship with her, this too could have an impact. So, clearly and explicitly emphasize that there would be no ill will if she turned your suggestion down, possibly in the same breath that you're establishing the importance of your friendship with her.
We've been friends for x months, and that friendship means a lot to me. I don't want to do anything to put that at risk, and if this suggestion makes you in any way uncomfortable please, turn it down - I'd never want to hurt you or make you feel bad.
After that, decidedly before you go into your request, give just a little bit of background about your current relationship.
You know that Girlfriend and I have been together for a while. She's always told me that I could sleep with other people. Well, her sex drive has recently plummeted, and she's suggested that I try to find another sex partner to make up for that.
At this point, keep a close eye on pFWB for any expressions of hurt, outrage, or betrayal. If you see any signs of any of these things, again, be prepared to full-stop and emphasize how glad you are for your friendship with her. Then, as you describe what you're hoping to see, describe the boundaries as well.
So I was wondering if you'd be interested in something like that? She's happy for me to engage with other people sexually, but she does ask that no feelings develop in the process. We wouldn't be going on dates or anything.
Again, monitor her responses carefully. Curious trepidation is okay, but if she's clearly trying to push past a major emotional issue just to get what she can get, full-stop.
If she seems suspicious or hesitant-but-not-hurting, offer to talk to your girlfriend with her, if your girlfriend would be okay with that. This can help to allay fears that you're actually cheating.
If she's responding positively or with interest, you can probably take it from there. Talk about all the things you'd often talk about with a potential sex partner - STD safety, favorite sexual acts, et cetera. Have fun, but remember your boundaries - and if you find yourself unable to keep from crossing them, realize that you have to exit the situation that makes that difficult in one way or another.