This is something you can do with, e.g. Fidelity's FidFolios, but those are paid for via an AUM fee.
Can you do the same here? That is choose an index, but seed it with some amount of shares that you already hold?
Another potential annoyance is filling out your tax return. Can you talk a bit about how that would work, with all the trades throughout the year you'll be doing?
I'd also love any more info you can provide on how exactly you do TLH. A factor model linear optimization problem is interesting! When I talked to my Fidelity advisor she pitched it as the pair-wise solution you mentioned, and gave an example of "sell Pepsi to buy Coke". But while the drinks are interchangeable, I'm not convinced the companies are! So I'm still a little hesitant on the idea of TLH at all.