I/Q sampling with just one ADC
Is this an entirely stupid idea?

No, but you've just came to the conclusion that instead of sampling complex, with Nyquist rate being the bandwidth, you should do twice as many samples.
That simply means you're not doing IQ sampling, but low-IF or direct-RF sampling.

Mix the signal with a higher intermediate frequency and filter it in such a way that successive samples are equivalent to the I, Q, -I, -Q

so, multiply the real and imaginary parts with 1, -1, 1, -1 … each , then sample them.
1, -1, 1, -1 is is a cosine at exactly half the sampling rate. What you're mathematically doing is multiplying your IQ signal with a cosine + j· sine, each of half the sampling rate. $\cos ft + j\sin ft = e^{jft}$, i.e. that's a simple frequency shift.
That simply shifts up your signal from $f \in [-b/2; +b/2]$ to $f \in [0; b]$. I.e. you've just put your $b$-Bandwidth signal on an intermediate frequency of $b/2$. That's a low-IF heterodyne receiver, which existed before IQ was even discovered.
So, clearly, you can do that. I'd avoid shifting to only b/2, use something slightly higher if your bandwidth and sample rate allows, and forget all you've learned about complex sampling – we're back in real domain, and you simply have to sample with a rate that's higher than twice the bandwidth.

Also: As much hype there is around the RP2040: There's nothing new about having two cortex-m0 cores. It's not an especially fast microcontroller or anything, it really doesn't open any &quot;new doors&quot;. The things that are nice about it are its flexible IO logic, but you need none of that.
In fact, these cores are ubiquitous, basically every semiconductor company sells microcontrollers based on that core. Although it's dual-core and clocked at a rather high rate, it's the last microcontroller that makes sense here: You need something that can process samples, not two high-clocked cortex-M0 cores that have no floating point and only rudimentary integer arithmetics. Get something with a Cortex-M4F instead. You might be getting a faster ADC, too.