There's a lot more here than meets the eye/ear, but I'll start with stating that the amp itself is the least likely culprit here...apologies for what is going to be a fairly long read/essay/thesis...
Electronics. An average '60s Sidewinder/Mudbucker reads 28-32K, which in itself is not a friendly environment for upper mids and highs, especially given the pickup location itself. Presuming that the Newport is wired the same way as the EB-0, there are two caps in the circuit: 0.33 and 0.1. The latter one needs to be removed for the bass to start breathing a little bit. It still won't be anyone's first choice for a Level 42 tribute band, but the sonic picture will improve markedly IMO/IME. See here:
Gibson EB0 bass circuitry and wiring information >> FlyGuitars
Output volume. I almost never play any Sidewinder-equipped bass with volume above 8, with 7 being my preferred choice 90% of the time. This is where the sound becomes more mellow and pleasant, at least to my ears. Do bear in mind that I prefer a clean sound, and am not into Bruce/Pappalardi/Fraser types of tone. Late Jim Dewar's "round but clean yet deep" sound with Robin Trower is much more my cup of tea.
Playing style. A *very* light right hand is a must.
Strings. No flats. Sorry TB. I've head good experiences with DR Hi-Beams (SS), Stringjoy and SIT NPS offerings. My current choice are La Bella White Nylons in 60-115 gauge. Obviously, YMMV and heavily so.
Cabinet. By and large far more important than the amp itself in my book. I don't think that the vintage Ampeg fridges pair well with Sidewinders whatsoever. The amp itself - when played through a different cab - should behave far more friendly. My experience with vintage cabs (Marshall/Orange/Sound City 4x12"/120W, Bassman 2x15" and 4x12") hasn't been all that great regardless of the amp(s) involved. Somewhat newer - let's say from late 70s on - cabinets tolerate the low end a lot better IME. Current "fridges" may be fine as well, I've never used one myself.
Amps. Your original question, half an hour later...

... I'm not keen on the vintage Bassman range but have recently played a session through a '66 blackface Dual Showman running through an Altec 421-loaded cab and it was a nice vintage sound, albeit on the quiet side. Having said that, most Trace Elliot and SWR "heavy iron" rigs will work fine as well. A friend of mine has an Eden setup - no clue on the models, sorry - and that one took the Sidewinder with no sighing either. I would *presume* that most newer amps and cabs - mid price tier and above - would be just fine. My own amp collection includes a circa '71 Carlsbro 100TC (valve/tube), Polytone 101 1x15" combo (SS) and SWR Studio 220 (hybrid) and all of them play well with the mighty Mudbucker, as does the '90s Trace Elliot stack in my club of choice.
Apologies for the long-winded post, hope it helped at least a bit.
My $0.02 only...