The question asks if it is okay to write "electrical vehicle charging apparatus (EVSE)" when the EVSE is an abbreviation for "electrical vehicle supply equipment".
The question overlooks one important point: a parenthesized text doesn't have to be the abbreviation of the noun that comes before it.
For example, consider:
The North American Telemark Organization (NATO), the nation's largest tele group, offers clinics, workshops, and adventure tours across the country.
(source: Skiing Vol. 52, Iss. 3 (1999); pg. 151, "Accessible adventures" by Susan Schinier, via COCA)
It's obvious that NATO in the parentheses refers to The North American Telemark Organization.
However, let's consider another example:
Left managment 1983 for public service: deputy assistant defense secretary (NATO), ambassador to Austria.
Forbes,Vol. 148 Issue 9, 1991 Forbes 400 p150, 61p, "Billionaires" (cover story) by Seneker, H., Lataniotis, D.
This time NATO is not the abbreviation of deputy assistant defense secretary, but it clarifies what deputy assistant defense secretary the article refers to.
Because @oerkelens mentioned that he found the text in a patent, I searched for the patent and found it. In the patent (US 2014/0114448 A1: Method and apparatus for sharing electric vehicle and electric appliance usage data), there is this text:
 The present invention relates generally to the field of electrical energy supply and usage and more particularly to a method and apparatus for sharing usage data from Electric Vehicles (EVs) and fixed appliances.
 [...] Temperature of the vehicle's interior can be used to engage a cooling system, while still connected to electric vehicle service equipment (EVSE, or colloquially, "EV charger") or otherwise, in advance of a User's return to the vehicle.
 230 improved appliance management system (here, an EVSE 140 and controller 250) (not shown here is that this controller can have access to similar meter 160 as in FIG. 1)
Later, in the claim section, they wrote:
17. A system for improved appliance management comprising:
an electric vehicle (EV) in data communication with a first server;
an electric vehicle charging apparatus (EVSE) controller also in data communication with said first server, said EVSE controller configured to control said EVSE;
It's quite clear that they did not use EVSE as an abbreviation for electric vehicle charging apparatus. However, "EV charger" or simply "charger" is used to refer to the same thing as electric vehicle service equipment (EVSE) in this patent, as stated in  and other items.
The patent uses the term EVSE, EV charger, appliance management system, etc. as defined in , , , and other related items in the patent. Also note that the patent title is about "method and apparatus" (which I believe is for some legal reasons), so naturally the term apparatus would appear in their claims.
With all that said, my opinion is: an electric vehicle charging apparatus (EVSE) controller is fine.