Current regulations in Europe require it to be CE certified since it has MCU with a clock-speed of >9kHz. So if it didn't have the ATTiny84, it would not even be required for certification.
This is nonsense. Anything containing electronics needs to be CE marked. What directives that apply is another story.
The <9 kHz restriction was for radio electronics that sorted under the now withdrawn R&TTE directive and referred to 9kHz radio wave frequency. Intentionally emitted - not MCU clock speeds etc. This scope of 9kHz - 3000GHz has, as far as I know, been withdrawn in the current RED directive. Currently it just says "anything below 3000GHz".
The first thing you need to figure out is what EU directives that apply to the product. If it contains anything radio/wireless, then it must conform to the RED directive, which also covers EMC. And if you need RED conformance, then implicitly you also need to low-voltage directive conformance (LVD). LVD in such a context is mostly about product safety, "not allowed to catch fire" etc. If it doesn't contain radio, then it must conform to the EMC directive instead. There's also the environment directives RoHS + WEEE. For products containing batteries, WEEE in particular is a bureaucratic nightmare.
From there, you have to figure out which normative standards under each directive that applies. This isn't easy, one typically has to consult an EMC/CE marking expert such as a test house.
Once you know all directives and standards, you can make a draft EC declaration of conformity listing them. If you have that, then there is a self-certification procedure which doesn't require testing at a 3rd party test house/notified body.
This self-certification procedure is outlined in the respective directive. You have to assemble a "technical file" of the product, which is basically just about producing a pile of more or less relevant documentation. For RoHS you must be able to point out the component manufacturer's documentation (datasheets etc) where they claim RoHS compliance, for every single component in the product.
The problem with self-certification is that if your product turns out to be non-compliant, you have vouched for it in your EC DoC and you'll be legally responsible. One good thing with hiring a test house for formal testing is that you can prove that you've done everything you can to ensure compliance, should you end up in court.
Basically, the EU bureaucracy system is rigged to make it hard for small start-ups to make it to market without taking short-cuts, as described here.
As for what testing you can do in-house, it is almost impossible to measure radiated emissions outside a lab, and it is definitely impossible to perform radiated susceptibility tests.
If you don't want a formal test rapport where the test house is stating product compliance, you can test these somewhat "cheap" (around 1k-2k€) through pre-compliance testing at a test house.