I'm doing full body training 3x a week. My problem is I feel I lack routine, and I would like to add some form of exercise on my rest day. How bad of an idea would it be to do planks on my rest days?
So planks may be good or bad for rest days but it depends on what type of core exercises you do on your workout days. Core is a particularly hard exercise to do more than flex as part of a workout. As a result it can be OK to do core multiple days in a row (but probably shouldn't be every day.)
What I would recommend is doing yoga which will dynamically support the core but will also support active recovery and flexibility.
Personally I would say not a "bad" idea, likely even a good idea especially if that's a skill you wish to get stronger at. When you say full body training is a bit vague, I have to assume that doesn't mean hitting targeted weighted core exercises then an hour of planks 3x a week. As mentioned above the core is used in pretty much every exercise but harder to specifically target to failure like you would other muscle groups.
I might be reading into it a bit much but it's interesting you feel you lack routine, workout 3x a week, and are looking to do more exercises on the other days. I'm not sure of your schedule but if you're looking to more and feel up to it I'm sure you could add planking or more to your week.
If you are worried about this extra work taking away from something I'd say make it a bit more structured. Say for the first few weeks do something like 3-5 sets of 1 min planks (more or less time depending on your current level). Then start to advance it like add in side planks, more time, leg raises during the plank, alternating straight and bent elbow planks, etc. That way you can tell if you are getting stronger at planking but also pay attention if you feel this extra work is taking away from your full workout days.
Now if you're looking for other things to do for active recovery days that's a whole other topic, you could do yoga (mentioned above), dynamic stretching, targeted rolling, lite cardio (like walking), working on exercises or areas you'd like to improve on (like planks or other body weight skills), a sport or activity you enjoy, etc.