I have the same sort of problem with my explorer. its been on a trickle charge all the time but when I pull the clutch and press start all I get is a click and nothing . Dash lights all on but wont start. Eventually starts after pushing start button dozens of times. Any ideas anyone?
If it does start eventually, I don't reckon its the battery. Seems unlikely to suddenly work after not working.
If there is always a click, it's not the starter button or wiring to it.
You seem to imply that the dash stays powered up while the button is pressed but it's not starting. I reckon that makes it solenoid contacts or starter motor brushes, or a shonky lead/contact to the starter. -Or something in the HT system.
Does rocking the bike against the engine before trying make any difference?
If you have a multimeter (and you should have), monitoring the battery voltage while trying to start it will rule in or out battery issues. Fortunately the TEx battery is easy to get to -just take off the seat.