It has long been the standard recommendation to not use rechargeable batteries in a weather station because of the normal chemical composition of rechargeable batteries have the limitation of only putting out 1.2V instead of 1.5V of regular Alkaline and Lithium batteries. Often the best recommendation was to just skip Alkaline for cold weather climates because only Lithium batteries can reliably work in freezing temperatures. Rechargeable batteries were typically NiCd and NiMH, both producing only 1.2V, which is short for devices designed to work better with 1.5V. For some applications 1.2V is okay, but it seems that for weather stations and radio transmission it was not a good solution.
There are some weather stations that don't have a super capacitor and are designed to be used with NiCd or NiMH rechargeables. These are likely now older or cheaper models. Modern decent weather stations at all price ranges have now adopted to using super capacitors and only relying on batteries for backup for low sunlight days.
But it seems we may need to change our standard messaging. There is now a new type of battery that is rechargeable and puts out 1.5V. From what I gather these started to become available mid 2019. Several companies are making these. The secret sauce is to use Lithium-Ion (Lithium polymer), not to be confused with non-rechargeable Lithium batteries, but in addition these new batteries have circuitry to reduce the normal voltage of a Lithium-Ion 3.7V down to 1.5V so that they can be made in traditional sizes of AAA, AA, C, D.
Has anyone tested these? They are very expensive. It probably isn't worth it for weather stations that only use batteries for backup for when the internal super capacitor is short for prolonged days without adequate sunlight.
BOTTOM LINE
Still it presents the situation that we can no longer say that rechargeable batteries provide too low of a charge at 1.2V, now that 1.5V rechargeables exist. Now we must clarify that not all rechargeables are equal. But using these new rechargeables is probably still not the best solution because of cost and because of unproven track record only because of newness. It is possible that these new batteries may be the way to go in the future if the costs come down or if proven to be a better environmental choice or both.
Here are some of these (they come in all sizes not just AA)
https://www.amazon.com/TENAVOLTS-Rechargeable-Batteries-Included-Electrical/dp/B07BKV793Phttps://www.amazon.com/Constant-1180mWh-Lithium-ion-Rechargeable-Batteries/dp/B07R1J6D3Zhttps://www.amazon.com/dp/B07MWYYDTM/https://www.amazon.com/dp/B07QW531W2/