Honestly, I'm not sure that you can trust her. When it comes down to a battle of wills like that, grandmas sometimes think they know better than you.
I'd just ask her straight out - are you going to keep him rf'ing in your vehicle and leave the seat the same way I install it? Do you promise? Do you understand what is at risk and why this isn't a choice but a requirement? (maybe show her one of the youtube videos?)
Open a dialog beyond the one you had today and see what her thoughts are. Get a feel for if she'll mumble under her breath every time she puts your ds in the car, but still do it properly and keep him rf'ing - or if she'll nod her head and agree and then do her own thing.
Ultimately, you have to decide whether or not you can trust her. You've had many years of experience in dealing with your mom and drawing on some past experiences might help in knowing if she'll respect your decision or not. You don't have to let her transport him, so if you don't trust her on this, then just don't allow her to transport him.
Right now it sounds like she just thinks "whatever, it's good enough" and doesn't understand the importance. Maybe dialog and an explanation - not overwhelming her or providing every link you possibly can, just a simple & straight forward explanation - like say the article that was published in the journal of injury prevention that gave statistics about how much safer rf'ing is for kids under age 2 in comparison to ff'ing for her to review on her own, and then watching the youtube video. Maybe the combination of the emotional and the research/factual/peer reviewed/data based report will help her understand and make her compliant with how your ds is to be transported. (There is a youtube video, not the importance of rf'ing, but one that is done by a sibling to the parent of his niece. It's done in love, and why he's concerned for his niece. Perhaps this tact might work better with your mom? I don't have it bookmarked, but maybe someone else does.)
Best of luck with the situation and the discussion. I hope that she is able to understand where you're coming from. Otherwise she'll just have to accept not driving him anywhere...
eta: I just looked at your siggy and see that your ds is 34mos old. How often does he ride with your mom? Depending on the degree of conflict & your own comfort zone, I'd consider a compromise with your mom - tell her that once he's 3 you'll allow him to ff in her vehicle. At 3yrs old and being over 30lbs, he's not unsafe ff'ing. Rf'ing is always safer, but it's not unsafe for him to ff. I'm not 100% certain that at his age and size an all or nothing position will serve everyone best... Obviously only you can evaluate that, but I'm just putting it out there. I'm not saying your mom is right by any means, I'm just saying that it might not be unreasonable to allow him to ff in her vehicle for periodic rides.