The turbidity caused by the common carp was limiting light penetration to lots of the bottom area of your pond that has extensive shallow area compared to the size of the deeper area. FA ALWAYS starts attached to the bottom or onto hard / firm surfaces. Since your pond has large amounts of shallow & lacks rooted vegetation it has lots and lots of potential to grow FA. If FA can’t get enough light it won’t grow especially when transparency is 6” as it was in your pond pre-carp die-off. If you were okay with the higher turbidity and fishing was acceptable then consider adding more common carp or koi to roil the sediments and cause turbidity. Carp or bottom rooting tilling action also interferes with establishment or attachment of FA filaments. Plus rooting can dislodge newly growing FA and plants.
You say after 50 carp died and “With the water clearing up, I got FA along the entire bottom and edges of my pond.” FA on entire bottom means the FA received enough light to grow even in deeper areas. Water currently is too clear for the your goal of low FA in your too shallow clearer water pond. If some carp survived the carp disease, give them some time and they will repopulate to create more turbid water conditions. Then you will have a better appreciation for "ol'bugle mouth".
If you gradually over time add common carp, make sure you have high numbers of LMB to eat carp recruitment, or your pond could go from 6”-10” transparency to a very turbid ‘hog wallow’. A slippery slope indeed. Annual bass harvest numbers can also help encourage or discourage survival of carp fingerlings - depending on harvest numbers.
More grass carp(GC) could help increase turbidity because when they are at high density and lack vegetable foods they will search sediments for invertebrates and any vegetation that exists. Some Pboss members have reported too many GC coincides with too much turbidity. I have also witnessed this situation.
Gizzard shad(GSD) are also known to roil the sediment searching for food items which some of it is new FA “sprouts” and other periphyton growth on the bottom. Both carp and GSD over winter and reproduce, so they as will tilapia, make bass forage. Remember - as the turbidity increases it is harder for bass to sight feed and predation pressure on small fish is often reduced that allows more small fish to grow and survive. Ideally you don’t want most of these to be small carp or GSD. Plus the bass in high turbidity might not grow as well compared to water with 16”-28” clarity. You have an existing LMB population so adding some GSD might be an option.
High numbers of tilapia in a longer water retention pond can also reduce water transparency by encouraging nutrient recycling causing phytoplankton growth. Although Phytoplankton struggles to bloom in a flow through pond. When FA is limiting tilapia by “working the sediments” for perphytic algae foods and some invertebrates can increase turbidity. Tilapia will eat some meat items to survive but meat is not a preferred food. The "beauty" of using tilapia is they die each year and YOU as manager not the fish decide how many tilapia to have next year.
Do I/we assume that all carp in the pond died due to ‘carp disease’ or just 50 and you still have some carp remaining?? I question all carp died due to “carp disease”, because normally when a disease attacks a fish specie population there are a few survivors due to natural innate immunity. It is sort of like TB, flu, and smallpox in humans, some manage to survive. If all carp, as you know it, died then I think the causative agent could have been copper poisoning during an algae control attempt or some other chemical. If all carp died the same day or two and not spread out over a week or two it again points to a toxicity issue not a disease. Who diagnosed carp disease? There is a viral disease associated with carp:https://thefishsite.com/disease-guide/spring-viraemia-of-carp
Carp tolerate low DO; so DO is not the cause unless numerous other larger fish also died. In a disease situation individuals usually die over the course of several days to 3+ weeks. Carp and minnows are much more sensitive to copper toxicity compared to sunfishes and bass. So it is relatively easy to kill carp and not other species depending on numerous variables at the time.
If the pond is basically a tub and has a long water retention time, as in no stream nor spring inflow, then pond dye can help limit light penetration and limit depth areas were FA can attach and grow. Dye use would also keep transparency a limiting factor for FA. Dye with turbidity does not need quite as much suspended solids (turbidity) to achieve your goal of light reduction & FA reduction. There are pond dyes available other than pure blue that might better help reduce FA attachment. Dye is not very practical when the spillway runs water constantly or often.
I would not depend on microbe bacteria nor aeration to directly reduce FA. Promotion of Bacteria for algae control is IMO mostly to encourage sales. Most all bacteria additions are for decomposition of organics not nutrient binding nor nutrient abatement. FA thrives on nutrients, which nutrients depends of the type of specie of FA. Best reliable bacteria brands indicate to kill the algae / weeds first then add bacteria for timely decomposition of residue. I know a lot of well aerated ponds that get literally covered with FA during summer. Thus good aeration does not reduce FA. Aeration could be helpful to your pond by helping keep roiled finely disturbed sediments in suspension thus helping increase turbidity.
No easy answers or quick fix for your FA situation due to the bottom design of your pond. IMO long term fix is some sort of a rebuild for the pond. Same size it or downsize it to fit your budget, either way, rebuild it with steep sloped shorelines and minimal shallow areas.
All weed type problems occur in shallow water not the deepest water unless deepest water is only 6 ft deep. If you are spending $1800 to $2200/year on algae control using tilapia, it will not take long to accumulate the price for rebuilding the pond, maybe smaller but with a properly shaped basin and appropriate water shed. A well sealed pond does not need a very big watershed to remain full. With a rebuilt pond you will not have the annual aggravation of FA. That stress relief is worth something if not a lot of $$.
Experimentation of using a combination of any of the above options could be helpful in reducing your FA problem.
Commentary – If one looks at this situation from the fishes’ standpoint., they “enjoy” and benefit from the big reduction of angling pressure during FA season. Focus most fishing and harvest when FA is wanting. This in turn improves catchability and less chance of producing hook smart fish, both good things.