What's the best way to split expenses with customers and bill them once a month?

Yes that does seem to be the cleanest. Negative markup certainly works. Do use double-sided items for these expense since 100% of the actual paid expense is yours and 50% comes in as income, not as contra-expense. If any of the expense payments are 1099 eligible you have to issue the 1099s for the entire amount you paid out and it is up to your customers to give you any applicable 1099 for what you billed them

Instead of "negative markup" you would be using Splits = enter the data in two or more lines to allocate, like this:

Airfare $300, split between you as one-third, and each of the other thirds billable to two different customers.

Travel expense $100 (yours)

Travel expense $100 billable to Smith

Travel expense $100 billable to Jones

and that would be done using a service item, set as "reimbursed = I incur it and charge it to my customer" so that you link Travel to your own Travel Expense account and to your own Revenue or income account.

That way, you see $300 total travel cost, and when you charge that to the two customers, you will see $200 total Income, which reflects the amounts you charged as "sales."