I think we need to rule out the room (standing waves) as the problem before we start recommending expensive pedals. I know this is talkbass but new gear isn't ALWAYS the answer.
Two questions:
1) Do these low end spikes happen on specific notes or all across the low frequency range?
2) Does the problem occur only in ONE specific room or does it also happen when/if you're playing gigs at different locations?
If you experience low end spikes on specific notes, and only in one specific room, your problem is most likely standing waves in the room. A compressor will not help with this at all. The spikes happen acoustically in the room so the compressor will never "see" these spikes.
A HPF might help a bit but if the spikes occur in the "upper bass" frequency range you will have to filter out a LOT more lows than you (probably) want to reduce the the spikes. The problem with HPFs is that they will increasingly filter out ALL frequencies below the cutoff point. They are NOT made for surgical cutting of problematic frequencies. You may end up ruining your tone without even getting completely rid of the spikes. HPFs are more suitable for filtering out sub-bass rumble.
The right way to deal with standing waves would be either acoustic treatment of the room (which can be very expensive, and in some rooms it won't even solve the problem) or get a parametric eq to zoom in on the problematic frequency(ies) and cut them. Simply lowering the bass knob on your amps may also help a bit.
If your spikes happen across a wider range of low notes, and in several different rooms/locations, we can start looking at other options.
All of the below will assume that standing waves have been ruled out as the problem (and you REALLY need to make sure this is the case before you spend money on expensive pedals).
A compressor will most likely not solve your problem, at least not by itself. If anything it can ruin your dynamics by overshooting on strong low end peaks. Many bass-specific compressors have a HPF in the side-chain, which means they won't overshoot on low notes but this also means they won't necessarily react to your low frequency spikes at all. The result will be unpredictable at best. If you get a compressor (which can be a good thing regardless of your problems) you should at least insert it AFTER an eq/filter that has already sorted out the low end peaks.
As for what type of filter/eq to use for this... it depends. A HPF might work but if your peaks are in the "upper bass" range you may have to set the cutoff frequency very high to affect the peaks at all. This can potentially ruin your tone. I generally don't like using HPFs with the cutoff frequency set higher than 40-60 Hz. Any higher than that will make your tone noticably thinner (which isn't ALWAYS a bad thing of course).
A shelving eq will be less intrusive than a HPF but will still enable you to reduce the lows and maybe reduce the peaks a bit. I'm not familiar with that Sunn amp but the bass knob on your Fender amp is basically a shelving eq. Have you tried to simply reduce the lows on your amp? Be aware that a (relatively) "flat" setting with a Fender tone stack means setting the bass+treble very low and the mids pretty much all the way up. If you have the bass knob on your Fender amp any higher than 2-3 you may want to try a lower setting before you buy expensive pedals to basically do the same.
You can also try adjusting the pickup height under your lower strings. The pickups on a Starfire basses aren't height adjustable but I believe the individual pole pieces are (I really should know this for certain as I own one but it's not in front of me right now).
And finally, without hearing you play we can't rule out the possibility that this is simply caused by poor playing technique, in which case the solution is PRACTICE.
