Other than having two schedule C's your tax situation is no different than if you had one sole proprietorship. You file one Schedule SE that combines the income from all Schedule C's and you report combined profit/loss on your 1040.
How it gets reported/divided across Schedule C's is not something I am incredibly confident of but I have seen similar situations with 1099's issued to one spouse instead of a joint business. My understanding is that in general you should match any received forms to the appropriate SSN/EIN. So, if the 1099-K was issued to a specific EIN then it should be reflected on the Schedule C for the business with that EIN, and you'd account for the portion that belongs to the other company by entering an item in Part V, Other Expenses.
The other expense entry would be something like:
Income assigned to Nominee - OtherBusinessName - OtherBusinessEIN"OtherBusinessEIN
Then you record that amount as revenue to your other Schedule C and you've got the 1099-K properly matched to the right Schedule C and you've accounted for the revenue that doesn't apply to that company.
If the 1099-K is issued to your SSN and your businesses have EIN's then the cleanest approach might be a third Schedule C with multiple income assignments.
Hopefully others weigh in if anything I've stated is flawed. If you haven't yet set up your businesses there are a number of good articles on how to organize multiple businesses under one parent/holding company that should help. Also, as others mentioned, understand the terms of your merchant agreement(s) to see if this is even permissible.