I'm getting some equipment together for my first attempt at AG. In fact, I'm more or less done

I have access to a system which is basically 2 large Stainless Steel pots, on the same level with the ability to pump water from the bottom of one into the open top of the other. The bottom taps can be turned on/off individually, as can the top tap which is on a swivel (as in a kitchen tap) meaning I can transfer liquid both ways.
Sort of like this: (!)
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Each has a burner and the right-hand one has a temperature control.
My idea was to use the left-hand pot (no temperature control) as my HLT and then to pump into the right-hand pot with the temperature contol where I'd conduct my mash.
When done, I could then drain the mash tun and collect the runnings into a spare fermenting bin before using the remaining water in the HLT to batch sparge. Eventually I'd have a bin with my pre-boil volume which I could then transfer (pick it up and pour it) back into the HLT which would then double-up as my boiler.
I'd like to skip the step of having to collect into a spare bin if possible - so, would it be ok to leave the mash water in the tun, introduce my remaining water and circulate the whole lot from the bottom tap and back in through the top? Then pump it all back to the (now empty) HLT for boiling?
Would this be a no-sparge or half-arsed sparge?! Not conventional I know, just wondering if I'm missing a big reason for this not being a plausible idea.
Cheers for any thoughts...