The most sensible advice about exercise after giving birth comes from authoritative medical organizations such as this:
They state that in the past it was recommended that women wait for at least 6 weeks after vaginal birth to do any exercise. But the latest guidelines suggest that these women can begin exercise as soon as they feel they are ready. This is for "uncomplicated" vaginal births.
For Cesarian births or more complicated vaginal births (such as with an episiotomy), it requires a doctor's approval first. Run it by your doctor, telling him/her exactly what you want to do and if that would be a good or bad thing. The general recommendation is to avoid exercise for 6 to 8 weeks following a C-section:
Mostly, you're trying to avoid ripping your stitches or getting your wound sites infected when you have just had a C-section. Exercise may also have an effect on scarring, so you have to be careful not to pull at the incision site or else it will worsen your scars. And after C-sections, some women have reported that internally, the wound has caused an attachment of the wound site to internal tissues, which means that they feel pulling and pain internally when they twist themselves in different ways during exercise that they didn't used to prior to the C-section.
Mothers worry that exercise might have negative effects on their breast milk. According to the first article above, exercise generally has no effect on the quantity or nutritional quality of breast milk. But of course, there are outliers where some women report a big reduction in milk supply after a lot of exercise or after a large calorie deficit. So obviously, this is something you can monitor and adjust as you go, but no reason to avoid exercise in the first place.
There is a common myth that exercise sours breast milk due to lactic acid, but there doesn't appear to be any evidence to support this. See article: https://hygeiainc.com/common-breastfeeding-myths/
Breastfeeding mothers definitely need to make sure they're drinking enough fluids. This is something that is stressed a lot for all women who breastfeed, but especially those that exercise during this period of time. Otherwise it can leave you dehydrated and can decrease milk production.
One other thing that women generally don't realize until they actually start breastfeeding: Breastfeeding burns lots of calories. The body puts on fat during pregnancy, and it's to prepare the body for breastfeeding. All of that stored fat can easily be burned off during the first year of breastfeeding. So women need to be aware that when they're breastfeeding, they're already running a small calorie deficit on the order of about 500 calories per day, more or less. Adding exercise on top of that may greatly increase their calorie deficit which runs a risk of decreasing breast milk production in extreme cases. So it might require women to eat more than usual.
All women who have given birth (vaginal or C-section) realize that their uterus is still bleeding for several weeks after birth. This is because the placenta detaches from the wall of the uterus, leaving behind open blood vessels which continue to bleed until the body seals them off. During this period of time, women will need to line their underwear with post-pregnancy pads, which might restrict some movement or might make certain exercises awkward. It also means they should think about adding an iron supplement to their diet. This is usually in addition to continuing to take prenatal vitamins after giving birth (prenatal vitamins often lack iron and require an additional iron supplement).
As for doing karate specifically, I don't think anything is really a problem. Let her decide what she feels comfortable doing. She can very easily run through whatever kata she has, one after the next, trying each one out to determine what she doesn't feel comfortable doing. It's not a problem.
She may even find things like low squatting positions and longer stances and lunges will improve the muscles of her torso and pelvic floor, which are things she'll need to focus on after giving birth. In other words, there's no reason to stick with "safe" kata that have shorter movement and allow for higher stances. The lower stances and the longer movements are very beneficial post-birth.
Pretty much any karate kata will be useful for recovery after giving birth. As would a number of general core strength conditioning exercises such as crunches, planks, pull-ups, squats, and overhead presses.
Here are a couple of good articles on core strength training for mothers after giving birth:
Hope that helps.