I'm in the process of reading through this thread and haven't seen much posted about 7N7 tubes yet so I thought I'd chime in.
First let me say that my experience with 7N7's has been in a Schiit Saga preamp where the differences from one tube to another can be quite subtle.
Next thing: Regarding 7N7 to 6SN7 adapters - I picked one up from a Romanian seller on eBay for a very reasonable price and electrically it seems good to go (tested each pin to each socket - the pinout is correct and there are no pin-to-pin or socket-to-socket connections). I tested it in the Saga with a properly tested, strong RCA 7N7 and it worked as it should. This said, I began feeling some hesitation in continuing its use (I can find no way to open the adapter and I was having trouble feeling confident in the quality of work in assembly. In the meantime I found that Woo Audio makes a 7N7 to 6SN7 adapter. It's expensive at $40 plus shipping but, having been impressed with the sound of the RCA 7N7 in the Saga, I decided to purchase the Woo Audio adapter. Here's a link:
https://wooaudio.com/accessories/varioustubeadapters
I have a small collection of 6SN7's (a new production Russian made reissue Tungsol 6SN7GTB, a new production Russian made Electro-Harmonix 6SN7, a GE 6SN7GTB, several RCA 6SN7GTB's, a Sylvania tall bottle 6SN7GTA, and two RCA 7N7 local base - electrical equivalent of a RCA VT231 / 6SN7GT). I won't go into any comparisons from use in a Saga preamp as they may be of little value in a discussion about tube rolling in the Lyr 3 headphone amplifier. I will only say that the 7N7's are my favorite of the lot in the Saga.
It's reasonably fair to assume that the RCA 7N7's will be very similar the early 50's RCA 6SN7GT tubes (this is what I gather from what I've read on the topic to date).
I am waiting to take delivery of a new Lyr 3 this coming week. I'll post an update once I've had an opportunity to listen to each of these tubes in the Lyr 3.