Unless your mobo is somehow different than mine, the FLT on the PQFP 386DX is not connected to anything. I certainly do not have any "automatic" disabling with the PGA CPU installed - and frankly, how would that even work, there is no special signal for that on 132-pin PGA (unlike on the 486 PGA, to disable the on-board SX).
I had to do a mod, see that missing DIP14 space for different clock generator? There is a jumper, not populated, next to the two existing ones that set CPU clock frequency. It connects to the missing chip only so you can put a third jumper there, and wire one end to the FLT pin. The other end is already GND I think (can't remember now), if not then just make that connection too. Floated 386 still uses some power but not much, the reason it gets hot is being so close to the DLC which itself runs hot. Slap a small, quiet fan on the DLC and it will solve both of these issues (especially for 40MHz operation).
Surprisingly the DLC is usually able to over-power the PQFP 386DX and it kinda works even without the FLT being properly grounded but as you can see, only kinda. It will not be stable, not to mention bad for both CPU fighting.
One more thing, the only correct DLC cache setup that works in BIOS is to set A20 mask to enabled and flushing to BARB. A20M is connected, sort of, it's provided by the keyboard controller (and that's all it used for it seems, the actual gating is done inside chipset) so do not enable fast A20 gate option. If you set A20 mask to disabled BIOS will configure the CPU to not cache first 64k of each 1MB region, that's perfomance loss and the mask signal is connected after all. Nothing else is though so BARB is the only option (do remember to set RAM refresh to hidden). While it might seem like FLUSH works too, it's only because most of the time the small L1 is evicted before it has a chance to cause collision. But it can randomly corrupt CPU state. Well, if you are going to do FLT mod then you can also add hardware for FLUSH too - I plan on doing that.