Back2,
I have found the insecticides you mentioned (Rotenone/Pyrethrin) to work for a variety of bugs, & have not harmed my plants at all. Just be sure to mix it correctly (or slightly weak) as too much may be harmful. If you read labels, watch for advice like "use caution or not recommended... for sensitive/delicate plants like impatience, hibiscus, etc."
In my experience, Sarrs & Neps have been very "tough", and not sensitive, but you need to use common sense, such as not applying anything in the heat of the day in the hot sun!
I don't know that you need a systemic type insecticide, & while I have used it on occasion, I have not used that type on Sarrs/Neps enough to recommend it outright. Not that it has caused me a problem, but again, I have not used it more than a couple times, and lightly.
I would suggest uprooting one of the most infested specimens & wash it off, and examine the roots & plant closely. You should try to determine if the beetle/bug/ant-like savages is actually chewing, sucking or directly feeding on the plants or their roots. Sometimes bugs will cause problems by infesting a plant, but not directly causing them physical damage.
Try to determine what kind of damage the bugs are actually doing, and unless they are boring into the plant itself, a systemic may not even be needed. Also, it the bugs are eating the leaves but not the roots, then a systemic or spraying the leaves is in order. If however, they are gnawing on the roots, then soaking the soil & not spraying the leaves may be the safest route. Most of this is finding out what you are dealing with & then using common sense.
Posting a couple good pics may help people here determine what kind of bugs you have, which is important if you are using an insecticide (especially a stronger one). Some types only work on certain bugs, and are near useless on other types of bugs.
I have had many years of garden type experience... with many bugs & treatments, and have brought that experience over to the CP side of growing things. However, I am only experienced with things here in the mid-west, and don't know much about those monsterous bugs down there in the south!
The sooner you get on this the better... & I would also recommend treating a few plants to start. Again, avoid the heat of day, full sun, hot-dry conditions, etc. Treating in the early morning or late in the day is usually best.
By treating a few plants first, you should be able to quickly see if the chosen insecticide works, how well it works, and if it has a bad reaction to the plants. (What has worked for me may not work exactly the same for you! Any good grower knows this, especially with the variances of conditions & culprits!) I would also try to avoid getting any insecticide inside the pitchers!
I have used more potent & stronger insecticides on some plants, & they had very little negative impact on the plants, however I also drained & rinsed out the bog-pools the following day or 2 after treatment (using fresh water), for once the bugs were killed, there was no reason to chance exposing the plants to it longer than necessary.
Much of what people here are telling you is correct, however it is always going to be a risk, and it is going to be your risk. There is no way around that. Insecticides are not something we all regularly put on our CP's, unlike roses or lawns! So there is always going to be a bit more of a risk involved, for there isn't enough experimentation around to say for sure that you may not have any problems.
(Even though I haven't experienced any problems with the Rotenones/Pyrethrins or many other types on Sarrs & Neps, but then I use care in measuring, determine what insecticide is appropriate for the plant/pest (again, different pests often need different insecticides) and I use methods of care & application I acquired back when I was learning these things from some old time nurserymen.) And while I may use some things for Sarrs/Neps, I don't use the same things on everything (like dews & vft's).
I think people here need more info to give more accurate replies. [And then there are always those here with no experience... but lots of advice! They haven't appeared yet, but they will!] While you may not have gotten the answers you wanted, some people so far have offered some common sense thoughts & advice. But I realize you want & need more... but again, without some idea of what the bugs are or even what they look like.... along with what damage is actually taking place to the plants, even with actual experience with many insecticides, I can't offer you more advice.
You really can't expect anyone to be able to just give you the "right answer" & tell you the right thing to do... especially for a bug/pest that we haven't even identified yet!
BUT, to try to answer your main question outright... insecticidal soaps, Rotenone & Pyrethrins are about as safe as you can get. If you take a moment to google them, you will find that things like the Pyrethrins degrade quickly, and are even recommended for food crops.
I myself, valuing my plants, still wash off & change the water once the bugs are killed. But here again, just because they are safe to use on your plants, doesn't mean they kill every bug/pest that may infest your plants. You need to look at the situation with a little different & broader understanding.
Looking outside the box a bit, it is even possible that something like garlic/peppermint/soap or even mothballs may act as a safe to use repellent... driving the pests away. But again, I have no idea what pest you have (other than black bugs in the soil that look like sugar ants), nor has it been determined exactly what they are doing to your plants. If nesting in great numbers, their presence, soil movement & excretions may be enough to disturb the plants & roots. But there could be more to it.
Like most of the things here, one size doesn't fit all.
Good luck.