Basically, no. It seems like it is difficult to find information on this because of the different versions. I couldn't find any page ever mentioning Google Sheets API v2 or v1, so I'm going to believe only v3 and v4 exist. When this question was posted, I believe it was only v3.
The top answer gives a pretty good list of your options, so here I'm just trying ...
The domain alias is not the problem.
With Google the user account alias is the problem.
How Google choose to DKIM sign or not
For Google to add the DKIM signature the username must be one of the Google account aliases.
Say you have DomainA.com(primary) and DomainB.com(alias)
Your google account is my@DomainA.com
DKIM signature will be set when sending from ...