The bigger question is, can the stator keep up with the demand. I don't care how big a battery you want to lug around, if your "big thirsty fog lights" are using much more than 50-75 watts (amps x voltage, so about 5 amps) you'll always be draining the battery.

 I agree with Verslagen and would look into an LED alternative if possible.  I had PIAA but those might be overkill.

 I would rather reduce my power needs than increase how much power I have to haul around.

You could get a little less draw by disconnecting the front running lights from the front turn signals.
Just unplug the green wires from the front signals.
That will save about 10 or 15W constant draw.

We don’t have an alternator, we have a stator that produces AC that is rectified to DC. Perhaps Ricky Stators can build you a stator with a higher output. Or, perhaps your current stator isn’t making full power. Check the AC output of each of the three legs (the yellow wires) and make sure you are making at least 60VAC @ 3000 rpm and that all three legs are within a few volts of each other.

The stock electrical charging system only puts out about 100 watts.  The stock running lights and rear tail light are 8 watts each.....so 24 watts are used up there.  The stock H4 headlight is 55 watts on low beam and 60 watts on high beam.  The instrument panel has a small bulb is about 1.5 watts I believe.......and another 1.5 watts is added for the high beam indicator light when you are using the high beam.  All these steady "ON" lights use up 80.5 watts on low beam and 87 watts on high beam.  That leaves 13 - 19.5 watts to charge the battery and run the intermittent electrical devices like the turn signals, horn and brake light.  If you have a cell phone charger wired in......you have to add that into the calculations.

The battery and power can only make very "temporary" power supplies to make up for any shortages in power when you are using the turn signals and brake light....the stock brake light uses 24 watts of power - so when the brake light is on you are short 11-4.5 watts of power and the battery is discharging.  You recharge the battery during the period you are not using the brakes (or horn or turn signals).

Since you have changed to an LED headlight and running lights, your math will be a bit different.  You could disconnect your factory running lights in the front, as your fog lights are doing the job the running lights used to do (that will get you back 16 watts).  You could also change your tail light to an LED to get another 4 or 5 watts.