We have the standard Tesla Wall Connector (received free as part of a Powerwall 2 install), which was installed by Joju Solar, highly recommended them, they did a perfect install.
Though if you’ve got solar (larger than 4kW), may I recommend you look at the Zappi V2. The Tesla wall connector is dumb and can’t dynamically charge at the excess solar current you produce, which is usually exported to the grid. The Zappi V2 is also eligible for the government grant as it is smart.
Almost all home chargers are rated to charge up to a max of <32amps (7 kW). They are usually wired into your existing consumer unit, if there is space, and connected to a dedicated 32amp breaker. However, they can usually be reduced down to operate at <13amps (3 kW).
If you can check you’ve got a 100A fuse before your utility meter, you should be all good to contact an installer to do the job and should NOT need to contact Bulb. Joju asked us for a comprehensive set of photos of our utility meter, consumer unit and location where you wish to have the EV charger installed and will let you know if there are any issues.
As @0rinoco suggested, it would be more economical to move to a “Economy 7” style tariff. Bulb offers their smart tariff, with an off peak during the early morning. However, I recently moved away from Bulb and often pay less than 4p/kWh to charge and occasionally are paid to charge our EV