Ok now the isolator is either a diode block (3 large posts one small and cooling fins) or a Solenoid (2 large posts two small ones)
First case: Diode iosolator: A diode is like a check valve on a water line. Current flows one way (The arrow points towards NEGATIVE or in this case the batteries) and the short charger cannot push power into the chassis battery.> Thankfully these are kind of rare.. though I have used them.
For the Solenoid... That's not the part that controls itself.. There is some means of controlling the solenoid.. Some are controlled by the alternator (If it's outputting the solenoid pulls in and charges) Most today are controlled by a computer... If it's the ECC (Engine Control Computer) than no.. there is no reverse charge
If it's an Intelletec Battery Control system or other type of B.I.R.D. (Bidirectional Isolator Relay Device) then yes.
How to tell the difference.
After a period of NOT PLUGGED IN (even 30 minutes to an hour will do)
Method 1: Turn on lights..then plug in and if say 1-5 minutes the lights brighten (I mean VEHICLE lights by the way) It's Bi-Directional
Method 2: (Better) Hook up digital voltmeter to a "Vehicle" test point.. You may have a fuse block under the dash or the battery itself.. NOTE voltage (Should be around 12.5 or 12.6) Now plug in and come back and see if it has gone up.. NOTE you may have to wait a short time (like 30min to an hour).. If it's now 13+ you have a Bi-Directional Isolator.
Method 3... IF the 12 volt accessory outlet in the dash is a "Chassis" device (often is) and hot with ignition off... There are plug in voltage indicators Some are digital Some are Multiple light...
Again if shore power charges the chassis. they will show an INCREASE if you plug in 30 minutes after shutting engine off.
I will save you some search time on that last option
https://www.amazon.com/Zeltauto-Cigarette-Lighter-Digital-Voltmeter/dp/B00VL9JZ0K/ref=asc_df_B00VL9JZ0K/?tag=hyprod-20&linkCode=df0&hvadid=312064685033&hvpos=&hvnetw=g&hvrand=15882270661183137423&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9017021&hvtargid=pla-568712381669&psc=1
Home was where I park it. but alas the.
2005 Damon Intruder 377 Alas declared a total loss
after a semi "nicked" it. Still have the radios
Kenwood TS-2000, ICOM ID-5100, ID-51A+2, ID-880 REF030C most times