Your problem with lack of volume leveling is the bane of my existence. I live in an apartment with thin walls and noise-phobic neighbors. I often have to simply disconnect my sub woofer.
I don't buy TV's that don't have a line out port separate from the headphone jack. If it doesn't have a headphone jack, I make sure the analog audio output at least supports PCM pass through surround sound or it's RCA-out. I bought an Insignia TV with only a headphone jack and my sound through my Philips RCA-only receiver is terrible. It's much better on my Vizio TV that has RCA-out.
The thing I have to do with my thin walls to watch high quality movies is turn off all surround sound or Dolby digital settings on my receiver. I keep it to 2.0 stereo. I also use an old program called "AC3 File" on my PC for its "DRC" normalization. Only MPC-HC or Windows Media Player/Center works with it though. In Plex Player I use Windows and sometimes Realtek settings to turn off all enhanced theater sound effects. (Plex normalization causes the server to transcode files, which opens Pandora's Box for me.) If using Plex for HDR, on a smart device instead of Windows, I simply can't use it during quiet hours. However, in Windows, I have found turning off the Bass Management effect keeps the Bass down. The Loudness leveling sometimes make it worse and sometimes better depending on the setup. Try both.
I have been able to get the uneven sound to stop, even with Sound Leveling turned off on my Vizio TV's settings (because it produces a strange dampening effect.) I had to play with my speaker/TV/PC settings until I found the right combo. In my speaker settings I turned off Dolby and Multi-channel (used Stereo instead.) I also turned the equalizer to "Classical" because all levels are flat -- or maybe you can turn your equalizer off? Try turning your Bass balance way down. If you have a Midrange setting turn that up while turning Treble down too.
I have read that some amps or sound systems have a "Loudness" setting that might help. Your setup seems like it might have this? Supposedly it was popular with older stereos. I haven't found any newer systems with this "knob" yet. I'm also tempted to try and find a hardware equalizer to attempt to isolate the frequencies that are most bothersome, but that won't help with commercials so it's only a partial solution.
As far as software settings, what's really needed is channel control. All channels other than the center channel need to be turned down. If the setup is 2.1, the left and right channels should be turned up while the LFE sub woofer channel is turned down. Some software players (MPC-HC) have settings for this and it can help but I haven't had complete luck with it. Plex doesn't have anything like this. If you come across something other than the "AC3 File" software that I mentioned earlier, PLEASE let me know!
I will say that I've been driven to download video files with the commercials removed because of this problem. It might be in the interest of advertisers to help technology developers overcome this, or just stop raising the volume of their commercials! This along with making commercials less repetitive and intrusive might get people to actually enjoy them.