I owe tax but my payment isn't allowed to be e-filed into after June 6, 2019 due to foreign earned income. Since I know what I owe, can I pay now and file later? I'd like to avoid paying interest.

Yes, I would file an extension and pay the balance due with your extension to avoid interest accruing on your balance due.  Working overseas does not prevent the IRS from calculating an underpayment penalty. 

Generally, the IRS requires you to make quarterly estimated tax payments for the current calendar tax year if both of the following apply:

• You expect to owe at least $1,000 in federal tax for the current tax filing year, after subtracting federal tax withholding and refundable credits
• You expect federal withholding and refundable credits to be less than the smaller of:
  • 90% of the tax to be shown on your current filing year’s federal tax return, or (80% for year 2018 taxes)
    
  • 100% of the tax shown on your prior tax year’s federal return (please note that this only applies if your prior year’s return covered 12 months (a full calendar year) – otherwise refer to 90% rule above only).