Hi
With the caveat of I'm not an expert, here's my advice!
The issue with the tweaked EQ on the 'base' HU is a bit of a pain really.
The standard 'base' system has no amp so the HU powers the speakers directly. It has a high level output and no RCA's. So you're starting with a high level speaker output with a custom EQ for the standard speakers.
You could feed this into an amp with a high level input and use that to power your system but the tweaked EQ will remain and you run the risk of it sounding bad. This would still be the case if you used an adaptor to convert the HU output to RCA so you would be no better off at that point.
The shops you have spoken to haven't really addressed this.
Like I've said some people have ignored this EQ issue, upgraded and been happy with the results.
For me I knew it would be an issue and the advice from the most of the longstanding audio experts on here has always been to rectify this before upgrading.
I suppose the first thing to give consideration to would be the Alpine upgrade.
The earlier kits deal with the EQ issue by having an amp that reversed or modified the EQ to suit the new speakers that come with the kit. I haven't heard anyone complain about the improvement these kits give over standard but the bar is pretty low to begin with.
The current Alpine kits require the HU to be recoded and there is debate as to whether this actually does anything as it's been reported the new HU's won't recode.
I've also read about a couple of people who've have issues with PDC and Gong volumes with the new kit which BMW can't sort out, but this might just be an isolated issue. The kits aren't really that cheap either.
When I approached my local dealer about recoding the HU to 'Hi Fi' in order to get it to output a flat signal they had not done it before, didn't know if it would work and wanted £100 to try it.
If you're someone who is accustomed to twin 10" subs in the boot I doubt the Alpine upgrade will float you're boat. You could start with that and build on it but I wouldn't recommend it as like I've said it's not cheap and the money could be better used elsewhere.
If to begin with you are looking to add a decent set of components, some SWS's under the seats and a trunk sub(s) you could do this with a four, five or six channel amp or a combination. Eg. a four channel powering the front components and under seats then a one or two channel for the trunk sub(s).
In answer to the cheapest way to deal with the tweaked EQ other than changing out your HU (bare in mind you will likely lose PDC, Gongs and bluetooth where fitted), sticking to equipment from the UK at least, I initially considered the JL Audio Cleansweep. Allot of people use this and there are lots of threads on it. There's also YouTube videos on how to set it up.
This will take the high level output from your HU and flatten it. I believe it remains high level so you would still need to feed it into an amp that takes a high level input but that's no problem.
It has no DSP function other than to normalise (flatten) the signal, that I'm aware of so you would use the crossovers of whatever amps you fit and the limited bass/treble of your HU to set up the system.
In the US there is a make of amps called Zapco which have a built in DSP but they're not sold here and I've never found any online.
That really leaves DSP's like the JBL MS8, Audison Bit One and Alpine PXE-H650. There are probably others but these are the main ones I've seen referred to.
Of these the MS8 in most people's opinion (including mine) ticks the most boxes and is supposed to be idiot proof in achieving the best sound with little or no knowledge. I've not seen a negative review of it.
If you were to use this for example with a 'Technic' harness you would plug the harness in behind the HU and run it's cables to the MS8 (more than likely in the boot). The harness has all the wires labelled and it's very easy to wire up.
Power and earth hook ups are exactly the same as you would for an amp.
The MS8 will be the heart of your system, take the tweaked EQ and not only flatten the signal but by using a binaural headset during the setup process create a new EQ based on your cars environment, set up your crossovers, add time alignment for the various seating positions and allow you to setup bass, mid range, treble, speaker levels, sub volume etc. It also has a fully customisable manual EQ for when you get clever.
It could, on it's own power your 'base' system as it is, you could just add some tweeters up front and your subs in the trunk with an additional amp. That I'm sure would be a dramatic improvement within budget with plenty of scope for expansion.
Some people have used it to power after market speakers though, it has 8 channels rated at 18 watts RMS at 4 ohms. I'm using it on my Vibe rears and centre channel and they sound great.
There's not enough to power the SWS's or the Focal's on its own but like I've said if you're adding trunk subs you could keep the under seats standard initially and run them as mid bass instead.
SWS's probably would be an overkill if you are running twin 10's as well.
I've read the standard under seats perform well as mid bass and that would be a big saving right there as SWS's aren't cheap.
