How about using DynDns to give a pseudo domain name to your changing public IP address.
There seems to be a general problem about using dyndns with cellular routers. To what extent this is country-specific I don't know, so circumstances in the US, for example, may be different.
My picture of the cellular IP network is that it works like a giant LAN, but I can imagine that technically this isn't an accurate description. In the UK (and other parts of Europe too I think) what seems to happen is that all cellular IP addresses are effectively dynamic private IP addresses and therefore routable only in the context of the internal network (internal in this sense meaning the network's entire cellular network or at least a limb of it). So when a cellular router accesses the Internet, it effectively does so via some process like NAT at the telco central gateway.
What this means is that while a device behind a cellular router can readily initiate a link to a remote server, the converse is not true - you cannot initiate the connection from outside to a device behind a cellular router even using dyndns (because the routing would need more information than the single IP address broadcast by the dyndns feature of the cellular router). This isn't a problem using a wx data source that only needs to pass data outbound, ie like a WeatherlinkIP working in default mode, but is not good news if you need to contact a logger from the outside eg to request a data download.
There is a way round this by acquiring a SIM with a static IP address - network connections then just work as normal, though there issues like extra latency to worry about. Certainly in the UK these SIMS are available, although not always easy to source - there seems to be just one company that specialises in providing this sort of service on a one-off basis. The monthly plan for the SIM typically works differently to a general cellular SIM in that you just get a modest monthly data bandwidth allowance and no call or text allowance. I think we're currently getting 200-300MB/month bandwidth for around $20.
NB I'm really not sure that the description of cellular network architecture is accurate, although the ultimate effect of not being able to initiate contact with a device behind a cellular router from the outside is what seems to happen in practice. Feel free to correct this description if you know better - this is another of those areas where getting solid information isn't always easy.