I've run press a lot and stamina is so important. Generally I want 12 available guys, I'll go with 11 if they've all got good stamina, but it's really hard to press with 10. St. Olaf is doing great, considering.
In addition to going slowdown, I'd put at least 20 minutes into conditioning into every player that has some room to grow stamina-wise. Then I'd put the starters at FF and the backups at GT. The problem with putting your starters at Getting Tired is that if there is a stretch with no stoppages of play, they'll get so worn out that they never really seem to recover. The starters will generally come back into a game as soon as they're ready, regardless of the back-ups' level of tiredness -- anyway, that seems to work for me.
I'd also go ahead and experiment with putting the walk-on in the depth chart. I had to do it once before and it really helped. Now, this was before the last game tweak when stamina was even more of an issue. He ended up playing only a few minutes a game and the negative effect seemed slight, but the positive to having fresher scholarship players was obvious. Again, this was before the 2/3 update when a player's fatigue level had a greater impact on his play, so I can't say for sure you'll have the same results, but it seems like it'd be worth a shot to me. Good luck.