"So here's the million dollar question: how do achieve it all (if not most of my goals) of punchy bass, sweet vocals, and resolving-ness without harshness?"
In your case, I'd start with a high quality tube preamp known to deliver the goods. In order of increasing preference, I'd recommend:
Transcendent Sound Grounded Grid
Modwright SWL 9.0 SE
No problems with bass response here, and tube rolling has well understood outcomes with the Modwrights.
"It only does a decent job on vocals and sounds very harsh on metal (ex. Metallica and Mastodon). The vocals don't sound as engaging as it could be and the metal guitars are so forward in the high mids that it's fatiguing to listen to."
Metal is harsh and fatiquing to listen to. On CD, LP or live. Your system just seems to be accurate.
I think along the lines of Wilsynet, but in a 180.
Amps add more tube flavor than preamps. You are describing the sonic character of your 555II preamp as the problem. I would change the preamp to a solid state that is known to have that solid bottom, yet smooth and accurate and let your amp give you the warmth & sweetness that you are used to.
Under $1000 I'd recommend:
Muse 3 Signature
Audio Research LS-9 (if using balanced)
You might find a Bryston BP25 at that price.
Kick these around in your search, good luck and I hope this helps.
But his amplifier is already a warm, sweet KT66 / EL-84 based push pull amplifier. So if it's still a bit harsh or etched, perhaps a tube preamp is in order.
As for the 55II being accurate and neutral, just because it's metallic and fatiguing does not mean it is more or less accurate than gear which is warm or euphonic. More likely it's on the lean and fatiguing side of neutral -- steely and metallic is also a kind of coloration.
The best gear that I've heard has great tone, sweet and extended highs, full of body and density, great low end responses, transparent and neutral, and utterly non-fatiguing. But to get all of these things in the same gear, this costs dearly.
For what it's worth, if you only have need of one input, your source has less than 1000 ohm output impedance and your amplifier has 50K ohm or more input impedance, then an LDR based preamp like the Lightspeed Attenuator or the Warpspeed would offer tremendous value for dollar. I have found these types of preamps incredibly transparent without a hint of fatigue (feel free to essage me privately if you're interested in the Warpspeed).
semi-related. Way back in time, I used to own a H-K Citation V tube amplifier (not exactly a mellow amplifier) with a Adcom 565 preamp. It was a terrible combination driving Snell J3s, sounding bright, fatiguing, and annoying. As already stated, I would look at the preamp first - I'm a big Threshold / Forte fan when it comes to used gear. I'm sure there are endless possibilities, but something with a nice FET circuit will work wonders - or a good tube unit.
I would also consider the LDR approach. While there are a number of kit and finished supplier choices, all with their own advocates, if you have more than one unbalanced source, there are multi-input versions of the Warpspeed LDR. Although single input, only the also Warpspeed has a balanced version. If you only need an unbalanced single source, any of the alternatives should work as long as your equipment matches up with the I/O impedances of your LDR choice; i.e., verify with the supplier.
Probably not what you want to hear but your system will shine with small-scale music, not metal or classical rock. For that you need some horns, especially with a nice little amp. Preferably horns with a fairly linear impedance.
If sources are all digital, consider DAC with volume control into power amp into some horns, without a preamp.
Massive solid state power will give you more output before clipping with tighter bass and more freedom with speaker choices.
Not sure if this would be satisfactory with small-scale music (depends on your choices) but if not consider having multiple components to slip in once in awhile as needed. I prefer having a system built for large-scale stuff and using it for the smaller-scale stuff too but YMMV.