Need advice: 93 Cobra audio upgrade with no permanent modifications

So you have components in the front with a passive crossover so the tweeters and mids are separated and there is no frequency overlap with the exception of a few hertz as all passive crossovers do. Powering this with the factory amp is going to net you with the sound you are experiencing which is lackluster at best with zero mid bass. I would think that the rear speakers are under performing due to the sensitivity of the speakers being 90 dB @ 1W/1M. Typically you are looking for something in the 87 dB @ 1W/1M or lower if you can when just replacing factory speakers. The components up front are at 87 dB @ 1W/1M but they are looking for a lot more power than the factory amp can provide. My guess is the 80 watts the factory amp put out was at higher levels of distortion which makes it harder for the speakers to perform. I only say this from past experience but a lower watt quality aftermarket amp will outperform the stock amp by leaps and bounds. So you are battling two things with the factory amp (being like most factory amps) being a pile of under whelming crap and not able to power the front components correctly and the rear speakers being too high in sensitivity and also needing more power (RMS not peak). If it were me I would ditch the 3-ways in the rear and go to two-ways with a lower sensitivity like the R168X2's. They will just perform better overall and you really just want the rear to be fill and the fronts to be the beans of the system.

With all of that you are still dealing with the stock head unit and amp of which both are inferior by leaps and bounds to an aftermarket head unit in both sound quality and power. You have a couple of options which are:

1) install an aftermarket head unit, bypass the factory amp, and install an aftermarket amp
2) keep the factory head unit and amp then either install line out converters (LOC's) on the speaker output wires on the factory amp to get RCA signal or if your aftermarket amp has high level inputs wire it direct to the factory amp speaker outputs. The issue with this either way is making sure the LOC's or the amp can take the power level the amp will put into them.
3) install a non-premium head unit and then either high level into an aftermarket amp or use LOC's as the power output on the non-premium head units is low enough not to worry.

Another possible way would be to see if an adapter was ever made to get RCA signal out of a premium head unit. Possibly the guy that did the blue tooth conversion would know and it would not surprise me if Mustang5L5 would know as he is pretty savey with these stock head units.

If it were my car I would remove the stock head unit and amp, install an aftermarket head unit (do not use the kit that requires you to put screw holes in the plastic part of the dash...its a Cobra after all), and then install an aftermarket amp. I would take it one further and run all new oxygen free 16 ga speaker wire to the rear speakers and to the crossover for the front speakers. 16 ga from the crossover to the door speakers and 18 ga from the crossover to the 1" soft domes in the dash. All the factory wires are preserved and this will sound the absolute best. You should really look into a single 10" or 12" sub to complete the system but that is just me.

I've done stereo systems in 5 cars over the years, including my 89 vert, and I can tell you it's a challenge at best to get good sound out of an early 90s car without butchering it. My suggestion even though it would be more work is to just unplug all your factory wiring and run all new wires. You can run new speaker wire which will give you better sound, and just run power wires for the radio directly to t-taps on existing switched/constant power wires. As long as you don't have to cut any of the dash for the radio you choose, you can leave the old wiring in place and run a parallel system that can be completely removed without a trace. FYI a car this small doesn't really need an external amp for great sound as long as your head unit is good quality. I personally like Kenwood Excelon.

I put a system in my 89 GT that can be removed and nobody will know. Nothing was drilled and no wiring was cut, well other than the plug from the factory amp but it was dead anyway. I have kicker separates in the front. Mids in the doors with some killmat and the dash has the tweeters in the 3.5" location as mentioned above. Rear has some used kicker 5x7's in the factory location. I have two amps, one under each front seat. One powers all the speakers the other powers the 10" kicker sub I have in the hatch. I used the factory wiring with the appropriate metra wiring plugs harnesses for the radio with exception for the one for premium sound. I cut the mating connector off the amp and used that for the speakers instead of purchasing the metra one. It was dead anyway and I have another good working one so it was sacrificed. Speakers, got the metra adapters that plugged into the factory plugs. For a deck, I am using a used bluetooth Kenwood head unit that a friend gave me, no CD player so it is compact and light. Console is undrilled or modified. Other than the front separates the metra adapters, and misc wires, I had everything else kicking around. I am quite happy with the way it sounds. The only thing I am going to do, and I was looking at them earlier this evening, is to add a volume knob for the sub amp. I want to be able to adjust the amount of bass without having to go into the settings of the head unit or under the seat to the amp. I stream music 100% of the time and the gain difference from song to song seems to change drastically. Not sure where I will put it yet but I am thinking to hide it between the seat. Here are a couple pics that I had on my phone of my install.