Hi there Jcrous, probably too late for you now, as it seems like your system will work fine

but in case anyone else wants a super simple and cheap coolant system, this one way of doing it

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A friend of mine has a plastic water tank mounted above his lathe on the wall, it will hold say 5 gallons maybe more ?, with an outlet at the bottom with flexible hose going into a normal coolant adjustable pipe set up with a tap and nozzle on the lathe. He fills the tank with cutting oil and it drains by gravity through the hose and nozzle as needed once the tap is opened, it then drains into the lathe suds tray then drains from here into a further plastic container under the lathe , when this is getting full , hence the top tank getting empty, he pumps it back up to the top tank with a small pump from the £1 shop, the type that you attach a hose to to the inlet and outlet and has a drive shaft that an electric or battery drill chucks onto and rotates the pump.
He has used it for years and gives enough flow to do the job , plus there is almost nothing to go wrong

Cheers Mick