I'm filing as HOH as a resident alien, but cannot e-file because my spouse is NRA?

See this TurboTax support FAQ for filing when your spouse is a nonresident alein - https://ttlc.intuit.com/turbotax-support/en-us/help-article/tax-credits-deductions/file-taxes-spouse-nonresident-alien/L4AuRKrMd_US_en_US

Yes, perform these steps to see if you are able to electronically file. 

1. Prepare your return in TurboTax.
2. On the screen for your spouse’s information enter a name and date of birth. Leave the Social Security number field blank. Select Continue.
3. On the screen Tell us a bit more, check the box My spouse was a nonresident alien at any time during 2024. Select Continue.
4. Continue through the screens bypassing any messages or warnings about the absence of the spouse social security number.
5. Indicate that you would like to e-file your return and follow the instructions in the program to register or pay and transmit your return.

If your return will not transmit, or is rejected, return and follow these additional steps:

1. When you get to the “Let’s get ready to e-file” screen, select File by Mail.
2. TurboTax will continue to give you an error regarding the missing Social Security Number or ITIN for your spouse. Proceed to print with the error but write Nonresident Alien (or the abbreviation “NRA”) in the space for your spouse’s Social Security number before mailing your return.
3. Mail the return to the address on the instruction sheet that prints with your return.

Unfortunately I still cannot e-file. Isn't there a way to bypass this? I reside in a foreign country now so paper filling will be a big trouble.

No. If this method doesn't work, then you will need to print and mail in the return. The issue is strictly the absence of the ITIN or SSN.