It used to be possible to get dimmers with both a fixed and variable output. I.e. there is a tap-off between the 3-way switch and the dimmer electronics, that responds to the switch but not the dimmer level. If you can find one of these, the fixed output could feed the fluorescent while the variable output feeds the LEDs.
Alternatively, if you have space, you could use a regular dimmer control independent of the 3-way switching, i.e. just added inline with the LED while retaining the existing 3-way switch and its connection to the fluorescent.
The best method would be to upgrade both ends of the 3-way with a smart dimmer at box 2 and a remote control for it at box 1. Instead of two travelers, this normally only needs one and a permanent hot at both ends. Therefore what is now the black traveler gets re-connected as a hot feed, which is then available at box 2 to power the fluorescent via its own switch instead of through the 3-way.
Re terminology, each language and design standard has its own quirks. Some terms make more sense in one language, some in others, some are modern and some are antiquated, just like the designs of fittings. Is the uniquely British term 'consumer unit' actually meaningful if you don't know what it is? The US-English equivalent 'electrical panel' is much more sensible.