As FaR aS tHe BasS mAKINg tHe pRoCeSsOr JumP,

there are a couple of potential solutions:
Mind you, I'm still unfamiliar with the One, so you may have to translate these ideas into their settings.
Let's start at the encoder and work backwards. There may be a final clipper that might be getting hit a bit hard. Chances are any final peak clipper will be broadband just for absolute loudness, and backing off what you're feeding it MIGHT help... although usually broadband clippers chew up highs more than lows... plus, your signal really doesn't sound clipped.... doesn't look clipped on the Breakaway's scope, either...
Now let's back up to the compressor / density portion of the processor. First, does the processor give you the option to adjust how linked the bands are? If you can run them more independently (or completely independently; I'm an extremist), the bass should stop "pumping" as much. Second, how high is the input and / or the output of the bass band of the compressor? This could also be a problem area. Maybe it's clean going in but at too high a level creating greater density... or perhaps it's just fine going in but gets a bass boost going out to the clipper, resulting in the pumping there.
Often any multiband AGC will have the "link" and "level" options mentioned above.
Finally, look again at any compression ratios and input levels from the front to the end of the box. Ideally (in my book, at least) you want a very gentle AGC to start, then a "density" section of multiband compression, then multiband limiting to catch the peaks, and then a final broadband peak limiter to catch only the highest of peaks (in other words almost never activated).
One more possible source of EQ issues could be the board itself. You may be able to lower any bass boost in the board and then speed UP the bass compressor and get similar results.
Personally, I really like your current sound. Yes, it sounds like you are using the "smiley face" EQ curve (with boosted bass and boosted highs) but for what you're playing it sounds right.
I heard something strange at the beginning of Nicki Minaj's "Starships" that I'm suspecting was ultra low bass driving everything else down in volume... that really sounded like it was the compressor or, more likely, the AGC.
Two possible solutions:
#1, a high-pass filter. It may even be built in the One. Try even as high as 50 Hz and if anybody actually complains about the missing low bass (unlikely, and your peak right now is at 60 Hz anyway), drop it down to 35 Hz. That MIGHT help.
#2, if you have the option to put one more piece of software processing in front of the One, consider grabbing Stereo Tool
http://www.stereotool.com/ and turn on ***ONLY ONLY ONLY*** the "Multiband Compressor / Limiter section. Set max output level for all to 10.00, up speed all to .00001, down speed all to .0001, clipping all off, EQ off. Set compress / limit to all compress... Then set the pre-amp input so that the input meter in the main window NEVER turns red.
This will "dynamically eq" every signal coming in BEFORE it hits your broadband AGC. If anything is set faster, or anything else is turned on, it will be too much (and if you turn on certain features, they would cost you money). On my station, I've had ballads where the vocals were totally ducked and burred when the band kicked in... this fixed that problem once and for all.
If you have any money to work with, I'd see about getting a multiband to put in front of the One... I thought the one everybody likes was called "Airianne" by Wheatstone, but I can't find any trace... it was a 30-band AGC / compressor. Can somebody else fill in this detail for me?
Really, though, the One should be all you need; and personally, I would be THRILLED with your current sound. I'd back the bass boost off a touch and maybe see if there's a high-pass filter that you could switch on to dump the ultra-low bass your transmitter may or not be passing, much less likely your listener's radios are picking up.
QUICK HISTORY LESSON: The Orban Optimod started out broadband, but quickly turned into a "broadband+"... The main AGC handled all of the audio... down to 200 Hz. A secondary AGC tracked exactly with the main AGC UNTIL you got a big bass burst. Then the secondary AGC would "unlink" and duck the bass to avoid getting "holes punched" in your sound, just like you're describing.
Good continued luck... and GREAT job!!! The stream sounds really hot.
EDIT: After listening for... an hour? maybe, that bass is beginning to fatigue me. I actually reached up to turn DOWN the volume a little... and I like the song that's playing. Not good. Just a db or two down should do big, good things for your signal. Food for thought, anyway.