The answer to your question is "not really".
As you mentioned in the question, Maillard reaction begins with carbohydrates and aminoacids. Meat is rich in aminoacids, so you don't need to add it to your recipe, it is already there.
If you want to increase the Maillard reaction, you need to use a marinade that is rich in carbohydrates, the simpler the better. This is the reason why sugar, honey, syrup, apples, orange juice and similar ingredients are very popular in meat marinades.
Adding dairy to your marinade you increase the aminoacid level on something that is not the meat, so you add "competition" on substrate level on your Maillard reaction. It also adds more "incorporated" liquid to your meat, making it more likely to cook than sear.
The other answer mentions upping the pH with antiacid or lye water, I recommend doing this with caution, especially if we're talking about a meat that contains a lot of fat. Lye water is very basic and it is also used with lard or oil to produce soap. (Which is the degradation of the fat in fatty acid + glicerol and then the formation of a salt with the fatty acid and the base)