Looking for 5AR4 rectifier tube most like solid state

I have had good luck with Cryotone 5AR4 tubes. They match your description nicely - fast dynamic etc.  Unfortunately, they do not ship things out very rapidly and they are a little pricey. . 

EML 5U4G usually can be used in place of a 5AR4 and is very, very linear. You may want to look into that!

@boxsoft

That preamp should put nearly a constant load on that rectifier. So Voltage sag really isn't a concern unless the rectifier is shot.

If the power supply is properly designed (bypassed), the character of the rectifier will be very heard to discern!

If this is an older preamp it might simply be that some filter capacitors in the power supply have degraded.

The actual audio tubes in the signal path are far more important to how the preamp sounds!! What brands are in there right now?

I’m just re-reading and seeing this is about the SLP-05. I owned a similar unit with the ultimate upgrade and actually paired it with Ralph’s Atma-Sphere S-30 on a high efficiency system for a little while. Unfortunately, the preamp wasn’t a keeper for me due to the characteristics you describe. Regardless of the tubes and cables I put on the SLP-05, it felt slow and sluggish. It is colored is a very pleasing way, but I couldn’t live with what it did to the timing the way it over-colored the presentation. Best tubes I used with it to speed it up as much as possible were Sylvania VT-231 for all the 6SN7 and the EML 5U4G in rectifier position. 

Different take but you should investigate the Space Tech Labs STR. (super tube rectifier). I’ve heard it and it’s the real deal. The HIFI cave and the Decware forum talk about it with both giving very positive results.