Triggered by the comment Guido wrote in the KnightsQRSS-blog and with some experience in QSK-QRSS (see my ealier blabla on the QRSS-ified RockMite) combined with Colin's broad-A1A-QRSS the following idea evolved.
What about a "fast QSK", some would call it TDM, switching between RX and TX constantly in a comparably short period. What about going TX for 1/3sec and RX for 2/3sec. Obviously, some SNR will be lost for the RX and the average transmitted power will be just 30%. Additionally, the transmitted power would create sidesband, which, in this particular case, may even be wanted in order to add signature to the signal. In a regular receiver one would simply hear "blibb blibb blibb".
Speed: Since there will be only one blibb created per second, there would be a reason to slow down the message. QRSS10 would probably be the fastest one could go. I calculated that my relatively long ident (not full call!) would take about 6min 30sec. That would nicely fit into a 10min frame for averaging.
Averaging, speaking of it, could help for the receiving too. The trick could be to divide the 1sec period into three slots, in which two are reserved for receiving. Now, instead of transmitting constantly in one slot, the TX-slot could glide for every 10min cycle. Averaging the received spectra of three cycles would display the full 10min cycle.
Downside: I believe that not only the TXed signal will get some side-bands to it, but also the received signal will suffer from the same. However, it may be desirable to have those side-bands transmitted, for reception, those side-bands could render the spectra really messy.
For sure, there always will be the option of slowing the whole thing down to some WSPR-ish period, avoiding the side-bands.