I am a CFII and also a licensed attorney (although by way of full confession not in practice, having attended law school, passed the bar exam, and practiced only as a way to pass time decades ago waiting for airline hiring to resume).
There is something that specifies LSA in the Sport Pilot certificate issuance. The applicable FARs themselves specify exactly that.
I will happily change it if definitive and official information is promulgated (which I have not seen to date), but my personal view is that the OP, with only a Sport Pilot license, cannot LOG PIC time in the Archer, or anything other than an LSA.
My reasoning is that the Sport Pilot is not “rated” in anything other than ASEL LSA’s, as that term is used in FAR 61.51(e)(1)(i).
“Rating”, defined in FAR 1.1, “means a statement that, as a part of a certificate, sets forth special conditions, privileges, or limitations.”
A Sport Pilot license holder has an ASEL “rating” in LSA’s by a logbook endorsement only. The logbook ASEL endorsement is limited to LSA’s by FAR 61.303. FAR 61.303 states that a Sport Pilot license holder can operate only a “light sport aircraft for which you hold the endorsements required for its category and class.”
In other words, this is the “special condition, privilege, or limitation” placed on the Sport Pilot ASEL rating. A Sport Pilot is neither rated, nor does he/she have operating privileges in anything other than LSA’s, by express FAR language.
The Speranza and Herman letter rulings, while instructive on the distinction between acting as PIC and logging PIC time, are not applicable to this situation. As the earlier post pointed out, everyone involved in them was appropriately rated and had operating privileges in the airplane by virtue of holding a Private Pilot license with appropriate ratings for the aircraft.
In fact, the Speranza legal interpretation specifically states that an “appropriate aircraft rating” is required to log PIC time. A Sport Pilot ASEL rating specifically limited by FAR to LSA’s only is not an “appropriate” rating for anything other than an LSA.
These are always interesting discussions, best held over a cool one in a relaxing environment among friends after a good day of flying. But without something official from the FAA itself, all of this amounts to little more than what we Navy veterans used to call “sea lawyering.”
In this case the OP has a legitimate question on which there are reasonable arguments for both sides. And in any case it is mostly an academic discussion.
But express FAR language defines what a Sport Pilot is “rated” in and does actually put express LSA limitations on the Sport Pilot certificate issuance. And the “rating” is what controls on the issue of whether PIC time can be logged.