Issues with false bottom in beverage cooler

[Ayerate]

I've been gradually making the shift towards all-grain via BIAB partial mashes in my kettle, but I've been struggling with maintaining mash temps, so I wanted to get a cooler setup to make the switch over complete. OBK had their 11.7 gal coolers with false bottoms on sale so I picked one up.

Now then, here's the problem: that false bottom has about 1.5 gallons of space beneath it. I'm only brewing 5 gallon batches, so even with a mash thickness of 1.5 qt/lb on a 1.050 OG batch, I'm looking just under 3.5 gallons of strike water, so almost half of that water is sitting under the false bottom. Seems to me like the part above is going to be really thick. Is that gonna be an issue? And if it is, what should I do about it?

I have of course considered ditching the false bottom entirely and going with a bazooka filter, but I'd like to try and work it out with the false bottom. I could do a thinner mash, but it wouldn't take long before I end up with more than will fit in my 8 gal kettle after sparging. The other thing I've considered is maybe trying to fill some of that space beneath the false bottom to force the water up. I've heard of people using glass marbles to fill space in carboys to reduce the headspace in secondary fermentation, so maybe something like that? Thoughts?

EDIT:
Here's a link to the cooler, so you've got some visual reference for what I'm working with here.

 

[72Chevelle]

Could you fashion a small dip tube down for the False bottom? I was thinking about doing that for mine.

 

[Lefou]

1.5gal of space? That seems like a lot.

It's taken me almost two batches of All Grain to get my efficiency and grain bill calibrated with my particular mash tun and false bottom.
So far, I haven't exceeded 8lbs of grain and five gallons of water getting a 1.050 pre-boil gravity. My tun isn't the same as yours and I hope someone with a similar set-up can help out.