JOSAA counselling only grants admission to NITs/IITs/IIITs and GFTIs. Since the institutes that you have mentioned do not come under the following categories hence JOSAA counselling will not give you admission in AOT/Hooghly/WB or any other similar colleges.

No Parnab it is not mandatory for you to take part in JOSAA counselling if you want colleges other than NITs, IIITs or GFTIs. A candidate willing to get a seat in NITs or IIITs and have cleared JEE Mains are eligible for JoSAA counselling. You can go for West Bengal JEE counselling.