What can I add to a chain cleaning bottle to get between the links?
I am not sure that any of the additives that you mentioned are necessary. You already have fluid flushing through the open spaces of the chain. I'd recommend just changing your degreasing solution and shaking the chain again, several times if necessary.
In addition, dirt between the link plates may not matter as much. Most friction is generated between the rollers and pins. If you were looking to totally optimize your performance and chain life, you'd need to get the dirt and old lubricant flushed out of that area, not just between the link plates. Repeated washing in a bottle appears to be sufficient to remove most of the contaminants from between the rollers and pins (demonstrated below), and it should suffice to get dirt out from between the plates.

Parts of a chain

The image above is courtesy of the blog Bike Gremlin.

chain link
inner plates
roller
pin
outer plates

As I mentioned, the majority of chain friction occurs between the rollers (#3) and pins (#4). Also, our chains wear as dirt gets between these items. In conjunction with lubricant, this forms a grinding paste that erodes the rollers. Thus, you ultimately want to minimize the contaminants that get between the pins and rollers if possible.
Cleaning your chain
It is not necessary to remove your chain to clean it. Per this Cyclingtips article, most riders should find it sufficient to a) wipe their chains with a rag after each ride and b) periodically use either degreaser and brushes or an on-bike chain cleaner as described by @Mr_Thyroid in comments.
If you are determined to clean your chain more thoroughly than this, I'd recommend watching this video by Josh Poertner of Silca. He does demonstrate why a degreaser and brushes is inferior to shaking the chain in a bottle of degreaser. In his demonstration, he showed that 4 solvent changes in a bottle produced a very clean chain, which he described as nearly equal to an ultrasonic cleaner. Again, if you are determined to use a bottle and solvent, simply change the solvent as often as you feel necessary.
NB: He created this video as part of an instructional series for Silca's wax-based chain lube, which requires every bit of petroleum-based lubricant to be removed from the chain before application. You don't need to buy his wax lubricant. You should merely take his broader point about chain cleaning. My read of the video is that he didn't consider an ultrasonic cleaner vital for the purposes of applying wax lube. That implies that for general purpose cycling, multiple agitated solvent baths should be a very good solution.
Now, if you are deathly determined to extract every last bit of gunk out of the chain, by all means, consider buying an ultrasonic cleaner and finishing your chain cleaning in there. Remember that this is not necessary for almost all cyclists.

Some miscellaneous comments: new chains come in a cosmoline-like substance to protect them from corrosion. Some argue that this substance is an excellent lubricant, and should not be removed. Others argue that the factory lubricant is high-friction and is excellent at absorbing dirt from the environment. It strikes me that allowing the chain to soak for a few minutes could help dissolve the grease, if any is left on your chain. However, beware that many citrus-based solvents can cause embrittlement of the metal, as discussed toward the end of Poertner's video. Basically, do not soak your chain overnight in a citrus-based solvent.
Also, you may be aware than many quick links are not rated as reusable. Consider how much force you have to exert to close many 11s quick links. While many riders including myself have re-used our links without apparent issue, it is best practice not to do this if the link isn't designed for it. I would expect that the link gets weakened with each use. The more you clean your chain off the bike, the greater the likelihood of an unexpected failure.
To my knowledge, Yaban (YBN) rates their links as re-usable for 5 times, and Connex has no specific upper limit. Some KMC links are reusable, but some are not; check the package. I have personal experience with 11s Yaban and 10s Connex links, and I believe either would be a good purchase if you are set on frequent agitated solvent cleans.