First of all, I did connected the data logger to the battery terminal and let it run for few days, with the load connected, drawing around 350mA.
Here a graph, starting with the battery charged.
Once the battery was depleted, with the load connected there is no way to have it recharged enough.
The weather was good at least for the last two days, so plenty of sun, but even with the 3W solar panel the battery was unable to be charged enough.
This indicates anyway the need to disconnect the DC/DC converter in order to have the battery charged.