I don't know anything about that specific one, but whenever there's a display with buttons that turns on a controller, it pretty much always works the way I described, with the display MCU turning on some tiny little transistor inside the display that passes B+ to the lock wire that then powers up the controller's LVPS. If yours is different, I'd be surprised, but you'll find out when you start measuring things in the wiring between them.
There are systems where the whole thing is already powered on whenever it gets battery voltage, and turning on the display just turns on the display and communicates to the controller that this has happened, but those are not very common, usually higher-end prebuilt scooters, bikes, etc., with custom-designed systems specifically for each one.
It probably has a relay inside it that closes when the alarm is turned off and power is turned on, and opens when the alarm is on to disable the system. If so, you can use that relay to do this job. But without knowing which connections to make, and what voltages/currents the parts are rated for, it's a guess whether it will work at all, or how long it will work, or if anything will be damaged (in the alarm or on the scooter) by doing so.
Some things can be determined by tracing out visually and electrically inside the alarm itself, and drawing out all the circuitry you can trace to reverse-engineer it, if you feel like spending the time, but it's a lot of work. It's probably easier to find a different alarm that does have a wiring diagram and info.
Allaboutelectronics and allaboutcircuits are good places to start learning generally. There's also a 555 specific site with lots of stuff:
555 Timer Circuits and a site you can use to "test" different component choices
Visual 555 Timer Calculator
What exactly the key/remote does, you'd have to find out either experimentally or from it's instructions and datasheet, etc.
What you *need* it to do when you activate the unit is to (use a relay/etc to) connect B+ of the controller to whatever wire on the controller (lock, ksi, keyswitch, ignition, etc) accepts B+ to turn it on (assuming that's how your controller works, which you'll need to verify first).
To *also* turn on the display, you would have to add a device (555, transistor with RC circuit, etc) that when the system is activated holds the power button circuit closed for just the right amount of time to do this, then opens the circuit again. Easiest to do if this device is powered from the same wire the controller is activated from.
Alternately, you can simply use the device (555 etc) to turn the display on, and the display will turn the controller on. Depends on whether you need the display or not (or want it).
How exactly the device needs to close the power switch depends on the way the switch is wired into the system. Without knowing that, the easiest way is if the device uses a relay to close it, since it won't matter how it's wired--it just closes the relay contacts wired across the switch, so that the display sees the switch as closed (button pressed). It's likely the switch just grounds a pullup resistor on a signal to the MCU, in which case the actual 555 output pin itself could be used directly wired to that signal trace. In that case, one side of the switch will be ground, and the other some voltage between zero and 5v, whenever the system is powered on.
Well, you don't need to *add* a delay. The delay mentioned is already in your system, by however long it takes the display to power up. Ideally you'd not have any delay but you're stuck with that one.
If you *want* to add a delay, using a(nother) RF xcvr setup (in addition to your remote) just to create one is a pretty complicated way of doing it. I'm sure something simpler could be done if you need it.
Was planning on doing this display modification maybe next winter, but I got curious so it might happen sooner. I’ll keep researching and post back if I have follow up questions or if I’ve made any progress.
