That's the reason I don't like Pyramid Golds on long scale basses. The E-string gets too thumpy and "dead" and the tonal balance across the strings is very poor. I love Pyramid Golds on short scale though. In fact, they're my favourite short scale strings. Unlike most strings the Pyramid Golds were actually designed for short scale basses. Specifically for Höfner but they were also the strings used by several 60s San Fransisco bands on their Starfire basses (Grateful Dead, Jefferson Airplane...). The short scale sets are MUCH better balanced than the long scale sets but they still need a break-in period where E+A will mellow out sooner than D+G. Once they're broken in the E-string will sound more or less like the A-string on the long scale sets (and the rest of the strings will be much better balanced too).
I've read comments about the Pyramid Gold short scale sets being too short for a Starfire and that people had to custom order them directly from Pyramid but I use a regular off-the-shelf short scale set (40-100). They are a perfect fit for my Starfire II but it's VERY close. The silk on the E-string ends just barely short of the nut slot. I guess it's possible that some sets may have the silk into the nut slot but if that's the case you can simply remove a tiny bit of the silk.
Pyramid Golds strings will last FOREVER. In fact, they just get better with age. I've got a set that's at least 20 years old on an old Hagström short scale bass and it sounds better than ever.
You will of course need to give the bass a setup when changing to different gauge/type strings. If not you may end up with more or less what I'm hearing in your GHS clip: A wobbly, "out of tune" tone due to incorrect intonation (which to be honest is what I suspect may be your problem with the GHS flats).