I like your idea
I think your first suggestion would be appropriate:
A bit about site designs on SE
Custom badge logos fall under "site design", which for unsponsored sites, is not likely to happen for a site of our size a few more years. The table below may convince you of the reason for this (questions/day is a 14-day average):
|Site still without a "complete" design
|Questions/day on 1/11/2023
|8 years 10 months
|8 years 6 months
|5 years 8 months
Quantum computing is a "sponsored" site though, meaning that it got a head start with its site design, but this may have been done in a rush by Strangeworks so that William Hurley could advertise the site at the SXSW conference (see Why does the Quantum Computing site look different?). The other site that got a head start with their site design due to "sponsorship" was AI.SE and they also don't have custom badge logos.
Interestingly, Data Science SE recently got its design done in August 2022 but with no badge logos, and same with Islam.SE, and same with Arduino.SE and same with Mechanics.SE and same with German.SE and Philosophy.SE and French.SE and Spanish.SE in September 2022. To make things even more confusing, Chess.SE did get custom badge logos in April 2023, but Writing.SE did not and Gardening.SE did not. Also strange is that Etherium.SE got custom badges with their site design in July 2022 but Bitcoin.SE did not in September 2022. Quantitative Finance SE got a custom badge logo in March 2023 and Sound Design SE got one in December 2022.
I now doubt that our sponsorship means enough to SE for them to prioritize giving us a custom badge logo. If you want to shed light on the path towards getting a custom badge logo, I suggest to find out why all of the above-linked sites that did not get custom badge logos despite being older than QCSE (and in all cases except for Spanish.SE, also are more rich in terms of the total number of questions) did not get custom badge logos, or why Etherium.SE and Chess.SE did get custom badge logos, despite Chess.SE having fewer total questions than QCSE right now
More related information
In this Meta.SE answer we see that custom badge logos are "possible but not promised":