What will be the steady state Charge distribution of given configuration- battery vs capacitor

I asked this question on electrical engineering stack exchange but I didn’t get any reasonable answer ,so I think this question is better suited here.

I was try to figure it out what should be the charge distribution over the spherical conductor of radius R (in figure) in following two configurations-

Assumption- length of wire (l) connecting battery (capacitor) to spherical conductor is very very long , wire and battery are ideal and initially charge on spherical conductor is zero(before closing of switch)

1.in first case ,only one end of charged capacitor is connected to the spherical conductor

2.in second case, only one plate of battery is connected to spherical conductor

In first case(capacitor)- as soon as switch closes distribution of charges started until (sphere + plate of capacitor ) becomes equipotential and this distribution causes a New potential difference between the plates!

But when we apply same logic for second case (battery) – similar to above there would be a new potential difference between the plates but it contradict the fact that potential difference between the plates of an ideal battery is constant

On the other hand if we keep voltage difference between the plates of battery constant then it implies that there would be no charge distribution even if we closed the switch , but isn’t it again contradict the fact that conductor connected to same wire should be at constant potential (if current is Zero)?

Can anyone suggest how would distribution takes place in both cases at steady State?

Here is link of my question –

https://electronics.stackexchange.com/questions/552300/what-will-be-the-steady-state-charge-distribution-of-given-configuration-batter