Premiere Pro: How to sync speed ramped audio with video from linked After Effects comp?
Adobe Dynamic Link is a great tool, but unfortunately at the end of the day it can be problematic as it has its limitations. As After Effects is almost entirely focused on video, it doesn't surprise me that audio would be one of these limitations.
Before dynamic linking existed, we always had to do round-tripping (sending files to another program, working on it there, and then sending a new rendered copy back to replace in the original program's timeline). Visual effects work in the Adobe Suite that hits these limitations with linking, such as this sound-related one you've run into, just means that you have to resort to this old-fashioned method as you've found.
In fact because dynamic link can cause problems, if I'm doing anything other than basic text or whatnot I only use the &quot;Replace with AE...&quot; tool to send clips to AE. After it's sent over, I hit Ctrl+Z (undo) in Premiere to undo the timeline, breaking the link and getting back the original clip that I just sent in the proper place/time.
Therefore, I'm always exporting a fresh clip straight out of After Effects to drop into the Premiere timeline. There's a few reasons for this:

Dynamic linked clips in the Premiere timeline use a TON of processing power because it's doing the calculations for the dynamic linked clip on the fly, like when you play it back in After Effects without any ram preview cached. You've probably noticed that it can stutter and struggle to make it through a linked clip when you're playing through the timeline, and sometimes slow you down even when you're not looking at it because it's anticipating. For anything remotely intensive, I unlink after sending to AE to avoid this problem, and round trip it back.
Similarly, round-tripping also saves you tons of render time on the video when you're rendering out the Premiere file. Let's say you watch your final render and notice you misspelled something in the credits or a cut is off-time. Re-rendering means it's going to have to recalculate all of the dynamically linked clips again, whereas if you had your individually rendered VFX clips in the timeline, it can breeze through them as it's just a normal video file.
The dynamically linked clip is Premiere's interpretation of the After Effects file, and Tthis can cause some issues and visual glitches with dynamic link rendering just as can happen when dynamic rendering AE files in Media Encoder. These are usually really specific issues with transparency, color science, plugins, and even sometimes referring to the wrong composition or a seemingly old version.

I know it's quite annoying, but you've clearly gotten the basic concept of how to round-trip it without the link. Here's a few things that could help or improve your workflow:

Like you said, you can increase the AE comp time to give you more working room. However to avoid having to do it multiple times, I suggest you just add more time than you know you'll need, and use the &quot;trim comp&quot; tool at the end of your process. Drag the work area range to where your final length should be, right click it, and select trim comp to work area.
To expand on your point on organizing, my suggestion is to create a folder (in your OS explorer) parallel to your footage folder where you can save your new clips. Use a consistent naming convention so that you can always match it with the original clip. For instance, original file: A002_C003.mp4 should be saved as A002_C003-VFX001.mp4 or some other format which distinguishes it as the modified clip, and also allows for revision files to be created without having to go &quot;-VFX-ACTUALfinalfinal2,&quot; hahah.

Additionally, I'm sure you'll be able to find some other ways to shave time out of your workflow. If you haven't already, try to learn the After Effects hotkeys as it will save you a lot of time with even these steps you're describing.
I hope I helped at least somewhat! Sorry there's not a more convenient way, but I also recommend reporting this as a bug to Adobe, as it really should be a true functionality.