For those without the spare cash but some mechanical skills there is the "cantenna" which used to be well known a few years back.
If you search the web for cantenna multiple useful links will be returned.
For example:
So with the Ubiquiti NanoBeam M2-13, my connection now is pretty decent and meets my needs, although not quite as strong and robust as I would ideally like.
I suspect the improvement is mostly due to the overall improved antenna and signal processing of the NanoBeam compared to the TP-Link CPE210 I was trying with before. However, I have also been experimenting quite a lot with positioning and aiming the device, and this has likely also improved a little bit, so this is not a perfectly controlled comparison.
This connection is now stable for hours albeit at relatively low rates - typically MCS3 is the highest negotiated. The connection does occasionally briefly drop out, but so far this was only happened a couple times for a few seconds, and this is not problematic for my use case. I suspect a Ubiquiti PowerBeam M2-400 (or a MikroTik LHG 2) would indeed give me a better connection but, as I said, our landlord does not allow mounting a dish, so I have stayed away from those. (And hopefully the NanoBeam is inconspicuous enough that it does not count...)