The short answer is, YES.
As you seem to realize, there is no such thing as an STC for an experimental airplane, because the FAA does not issue a type certificate to begin with. Thus a supplemental type certificate would be meaningless, since there is nothing to supplement.
With an E-AB airplane a gross weight increase comes under the heading of a major modification. In all cases you should first look to your Operating Limitations for guidance about how to deal with a major modification. In all likelihood your OLs will require you to contact the local FSDO and ask for a flight test area and flight test period (number of hours) in which to test your major mod. You would then prepare new weight and balance calculations, re-enter the Phase I Flight Test period (as per the FSDO), fly off the hours, and make the appropriate logbook entries.
Of course, we hope that there is some reasonable justification for increasing your gross weight, but that is up to you to determine.