If you run slow enough (MAF or lower), you can concentrate on something else. Get a phone/tablet/TV with a Chromecast or Internet access, and get these YouTube videos going. Cat videos are probably fine, but educational ones are better. Have a backlog of videos related to your favorite sport to watch (technique, strategy, etc...)? That's the perfect time to watch them.
You can also use running on a treadmill as a meditation. Concentrate on what you do,etc... (So, the opposite of my first suggestion). Not sure this is the best idea though as my wife is a quite avid meditator (she can stay without moving or thinking for 2 hours and find that pleasurable), and she says that running on a treadmill doesn't work as a meditation for her. However, she is nursing an injury and has to be very careful, so she cannot let the body do its own thing for now.
I do Judo and Aikijujutsu (so I understand d the needs of a grappler). I'm in a dojo 5 days a week. Some things about my own training that I think you can use.
1) I've found I feel better if I do only Q&D sessions MWF by itself. If you add 5/3/1, then use the 2 day/week split on TuTh instead of after your Q&D session. Personally I've found GTG (Naked Warrior work) does nice, I use pullups and pistols (diff movements than Q&D).
2) Depending on your job do a nice light walk every lunch time. Don't worry about MAF, just walk and breath (or with breath holds). I work in a 6 story building, so when it rains I do stairs. If the heart is beating a little too much, just stop and lightly stretch in the stair case.
3) With the lockdowns it becomes more important than ever to maintain a schedule. Whatever your old BJJ class times were, keep that schedule and do solo work at home. It won't be as intense as class, but that's OK. There is always something you can do at home on your own to improve your body awareness for the mats. You might also add work that has nothing to do with BJJ, like a well controlled high kick. You won't have to shuffle your life around once things open again, and everyone around you will be used to your schedule...
As for the skip rope problem... One way is to just cut it down to 20 seconds on / 40-100 seconds fast&loose drills. Eventually you'll add more intensity or more time as you choose. You could also see what happens if you use jumping jacks or burpees instead of a skipping rope. An even slower more controlled movement.
+1 for the jumprope, but it does take TIME. I say stick with it. It seems to be much easier on the joints than running for me ( which I also hate) and I've made good progress over the last year. When I start to get a little short of breath I stop skipping and do some light goblet squats or farmer walks or just stretch a little until I feel ready to start skipping again. I don't have set intervals, I just try to keep moving for 30-40 minutes, avg HR sits nicely at MAF. Plus I really enjoy it for some reason. On other days I do either Q&D or barbell squats and presses, but I look forward to the rope sessions