If you split a 15 gallon batch into two 7.5 gallon batches, the 30lb grain alone will take up 2.34gallons of volume (30 * 10 / 128), as according to sizing the tun from HowToBrew, grain volume is 10 fluid oz per pound. Plugging in a 7.5 gallon batch at brew365, at a mash thickness of 1.5qt/lb, the water alone is 11.25 gallons. In order to get the mash to fit, you'd have to use a mash thickness of 1.0qt/lb. Then the mash water would be 7.5g and should fit with the 2.34g of grain, 9.84g. However, you will likely not get great efficiency at that mash thickness. It will also be difficult to stir without spilling, and difficult to stir effectively.
As you haven't tried this before, it's probably easiest to measure your first runnings, and calculate your total sparge water as
Total volume of sparge water = Target pre-boil volume - volume of first runnings
According to the previous calculator, your target pre-boil wort is 9.26g.
The available volume in your tun is roughly
10g(tun vol) - 9.84g(mash vol) + Xg(first running)
If your sparge water needed exceeds that, you'll need to do more than one batch, or fly sparge. Going back to the calculator, it estimates you'll need 6.66g of sparge water.
For sake of an example, lets say your first running is 4 gallons. Then 10 - 9.84 + 4 = 4.16g. If you split the sparge in half, and ran 3.33g, then another 3.33g, you should come close to your target.
I'm guessing you may be able to get away with two batch sparges, maybe 3, depending on your first runnings volume. If you're setup on both systems to fly sparge, you could do that as well.
One last concern is that if you are going to split the batch into 10 and 5 gallons for boiling, and 5 gallons 3 ways for fermenting, you will still need to somehow mix all the pre-boil wort together, otherwise it will be parti-gyle, where the gravity is different for each boil, which will produce different results. I'm not really sure how to accomplish this without using a ~20 gallon container.
My recommendation is to brew on the systems you have. If you have three, 5-gallon boil/fermentation setups, and two 5 gallon mash setups, I would recommend two parallel and one consecutive mash, into three kettles.