I believe the tube types that you listed have the following relative output level:
12AX7 = 100
5751 = 75
12AT7 = 60
12AU7 = 40
If the circuit was designed around the 5751 tube, I would start by trying different varieties of that type before jumping to the other types. The 5751 was specifically designed to US military specifications of ruggedness, low noise, and long life, at the beginning of the 1950s.
In my 25 years of tube-rolling experience, the absolute best 5751 tubes are the early GE and RCA with black plates and 3 mica spacers, rather than the later ones with gray plates and/or 2 mica spacers. These are commonly known as Triple Mica Blackplates. Older is typically better, with 1952 to 1955 being slightly better than 1956 to 1959, and day-and-night better than anything after 1959.
I have purchased, tested, listened to, and sold literally hundreds of these over the years, and only came across two of them that were bad. I currently maintain a stash of 40-50 of these, with an emphasis on the earliest versions with the silver "clips" or "braces" on the tips of the plates where they extend beyond the upper and lower micas, because I consider them to be the best-of-the-best.
Other 5751s to consider would be Sylvania Gold Brand or military triple mica blackplates, Raytheon with windmill getter, and maybe Tung-Sol. These are less common, usually more expensive, and IMHO, no better than the GE and RCA.. I have a very slight preference for the RCA, but the GE is much more readily available.
Google Joe's Tube Lore for a detailed comparison of all of the 5751 tubes, and other small signal tubes. A bit dated, but still quite relevant.
Feel free to PM me if you would like me to elaborate any more.
Let everything burn-in for 50-100 hours to establish a baseline before you start changing things around.
Enjoy the music!
Brady