First: Thanks to all for the input given!
To further dive down in the situation I want to dive in:
I want to both measure the behaviour of batteries whilst charging/discharging, and to control the behaviour of some budget chargers, and to be able to verify if some high-quality charger really reports to me the truth or what else it is doing...
As I also want to share those findings via a blog, I will need a practicable way of getting the measurements out of the test equipment. I already have an Agilent U1272A, that I got as a good deal on ebay, but still have to get the data cable u1173B- will also need that for firmware upgrade.
And as an exerienced computer guy, I know (and have kinda a hatred) against proprietary drivers I would need to get something going- which means that I would like to invest in some interfaces that still will be supported in the future and where a windows update will not break things- with my Agilent DMM, you have 2 versions of data cable: U1173A for Windows up to 7 and U1173B for Windows 10 and 8.1- Due to that considerations I would rather have my future equipment equipped with some classic RS232 and ASCII output or LAN with scpi/LXI. Or maybe Modbus, thats also some quite common and understood protocol...
For precision, yes- I have taken some time to look at shunt values in different multimeters and also looked at the higher ranges. I am yet in the process to define some requirements and test scenarios, so that the required ranges can be concluded upon.
Looking at standard NiMh cells with 1.2 V nominal voltage and the equipment they will power, the currents I will have to take into account probably will be between 100mA and 500mA. This can also be very well handled by a multimeter on its 3A range (as in my u1272), where the internal 0.01 Ohms shunt will be used then- with a display accuracy of +/- 1mA I am very satisfied, as the measuring accuracy in that range is about 0.3%- so it should give me in this regard a tolerance of overall around 1% of the measured value- thats something I can very well live with...
With Lithium batteries, as used in cameras oder portable lighting fixtures, like the Sony NP-F550 and clones, world is a bit different and possibly a bit easier, as the overall voltage is bigger. Here I also have to account for rapid power draws like that of a camera taking pictures and drawing for some Sec. some big power spike and then return to idle.
Fuji cameras are reported to draw 500 mA from their small internal battery in those scenarios.
Logging: Yes, ultimately I will want to log that data to a PC for further use. Due to some considerations (and to have an interesting journey, which also may be of interest for other folks) the OS on that box will be FreeBSD. Also I want to use mostly open source software in that regard.
(I am also pragmatic- my current box where I type this on, and do my photo editing consists of a Windows machine, and I use Photoshop and Lightroom and some other plugins...)
I have currently a deep look into sigrok, which looks promising, also will do some evaluation of lxi-tools in the next few weeks.
Budget: As this is currently a hobby, it may cost money... I think I am happy when I can put something around 1000-1500 €€€ over a time of maybe the next 4 months into equipment, also including some good quality tools and 4mm wires etc., to simply be able to build some adapters needed to accomplish the tasks I will want to do.
As a multimeter often is the centre of the bench, I would want to spend some reasonable money on that, simply because once you buy cheap, you buy twice. And I intend to use that stuff I will buy now over the next 10 years with good care- and when I determine to cease that hobby, I would want to get a decent price on selling them...
A look at equipment points in my opinion so Siglent- the SDM3045X would quite well fit into my picture, as well as the SPD1168X power supply. As I am not quite sure where this journey goes, I am a bit reluctant to jump initially on the electronic load Siglent provides due to its price tag, but this certainly is on my wishlist. Also a scope is on my list, as I will want to determine via a photodiode to measure the short burning duration of an electronic flash and PWM cycles of LED lights, because this is somewhat critical in some applications.
These have the advantage of being new and hopefully not needing repairs for some time- whereas some DMM like an Agilent 34401A or so, which goes on ebay for about the same price range, probably will need some cleaning and capacitor replacement in the near future...