Hello
it is absoultly a new model with the electric starter 06-08 black covers and
no kick start, implimented on your old bike, attach a wiring diagram of your KLX250E it has to be a DC wiring system to charge the battery, I had the same problem with mine KLX it was due a bad conection and a low power that gave a low strating RPM ther for in suficent spark,
so did a couple of things to fix it:
1. sand and clean all conectors, frame, battery, coil, engine mounting's.
2. got a new battery with more then 8 AMP'S it gave a nice RPM, the origenal is very low and runs out very fast, aspecely when the bike stands off for a long time.
3. conect a nice ground wire dirct to the earth lead of the stater/engine from the battery.
4. replace the spark plug, wiring and cup to the coil
5. and last check the coil ressistence I had mine replaced due bad isolation
hope this with halp.
good lack
:)

this is the 06-07 KLX250S wiring

this is the klx250D wiring I dont think this is yours it is AC system and can not work with a battery and a starter, just if add a rectifire befor charging, check on you bike!

last but not least this is the klx250E wiring as you manchend this is your bike with an 2007 engine so probebly aded a battery on conected to the main harnes brown for possitve (+) , black a yellow for negative (-)
E starter conected to battery with by a start botton and relay added.
not sure if you have all safty sircut's like sidestand switch, nutral switc, clutch switch, all of them can be bypassed for testing!

if you any qustion's dont hasitate to ask.
good lack