Humidity control in basements starts with controlling air-infiltration from the outside. Humid outdoor air getting into the basement is usually the largest source of basement humidity in the eastern 2/3 of the US. It's driven in part by stack-effect from combustion appliances (furnaces, gas hot water heaters, etc. but in multi-story homes it's the whole house height that becomes the stack. The biggest entry path is thin gaps in the band joists & sills, followed by clothes dryers (which you can't do much about.), then windows/doors. Foam sealing the band joist & sill is the first/best start. (Sealing the attic also reduces the infiltration pressures by a surprising amount as well, but for now we'll stick with the basement.)
When you're all done, test any combustion appliances for backdrafting while running the clothes dryer, air handler, exhuast fans anywhere in the house, with all doors & windows shut tight. If need be, add a purpose-specific combustion air supply to the room with the burner.
Then, treat ground moisture permeation paths- slab and walls can be treated with concrete sealers (radon sealer on the floor slab would be good, if it wasn't poured with an embedded polyethylene vapor retarder.) If you have a slab-depressurization radon system sealing the slab won't be necessary.
Before finishing the walls, putting up semi-permeable rigid foam insulation against the foundation (wiith foam/caulk sealed seams & edges) under any fiber insulation is advisable. This can be either extruded polystyrene (XPS) "pink board" or "blue board", expanded polystyrene (EPS) beadboard (like Styrofoam TM), or fiber-faced (but not foil-faced) polyisocyanurate aka "ISO". Foil facers will trap ground moisture in the insulation, may end up rotting out the foundation sill. In extreme cold climates vapor retarders & foil facings can be used (and are a benefit), but not for most of the lower 48 of the US.
Then, be sure to use high-permeability paints on the gypsum or you'll be looking at mold on the furring strips/studs in only a few years- it has to be able to dry toward the interior. Don't use vinyl or foil wallpapers- same story, only worse. There's a bunch of research & data on this- you can read up on it here:
Insulating your basement will make it warmer, lowering the relative humidity, but being warmer will increase the vapor pressure up from the slab (unless you insulate IT as well, which you might consider if you have the headroom. 1" XPS would give you R5, and much warmer floors. Standard density EPS and ISO don't have the compressive strength for use under floors, but 2lbs/ft^3 EPS does.)
I you set your humidistat to 60% RH a 1.5 ton unit should be able to keep it dry without overchilling the space, if you set it lower than 40% (or when running in thermostat mode) you may have a bit too much basement coolth at times, but worst case you could go back to the standalone dehumidifier and close off the register to the basement. You don't need to keep the basement bone-dry to avoid mold. At about 70% RH and up it can really take off over time, but at 60% you'll be fine. Could be you've been overdoing it a bit, but it could also be that by keeping the basement under 50% RH it's kept the rest of the house under 60%RH- it's been part of your whole house air conditioning all the time.
The low speed dehumidification mode is a great feature- at lower air speed it wrings more water out per cubic foot of air, which does indeed keep it from overchilling. If my (3x oversized) AC unit had that feature I'd be using it, but instead I keep the basement dehumidifier unit set to 60% RH year round, but this summer it's been high-humidity with low temps, and I've had to resort to running a half- ton window AC unit in an upstairs room as my whole house dehumidifier, leaving interior doors open so that all rooms get the benefit. If I ran the main compressor it'd get cold & clammy, not comfortable & dry. (I'm sure you'll be better off than me, but I'm not springing for a new AC unit any time soon...)