That's a good question, but not one that is easy to answer. I have, unfortunately, been involved with such problems.
One crucial thing to keep in mind is that councillors have an obligation to treat council officers (and in small councils that means the clerk) with respect. The clerk is employed by the council, and is not the servant of any individual councillor.
Ignoring these principles can lead the council into serious and costly difficulties. If the clerk feels that they are being treated unreasonably, they can make a formal complaint to the council. If that does not lead to a resolution, the problem could finish up with an employment tribunal, which can be very time consuming and can result in a substantial compensation award against the council.
Because of these possible consequences, a councillor who behaves in this way is acting to the detriment of the council, and could therefore be subject to a complaint to the local Monitoring Officer. Unfortunately, the enforcement regime is now practically toothless, leading at worst to naming and shaming.
Practical steps that may help involve other councillors taking action. It should be clear which councillors take immediate responsibility for employment matters, and it may be helpful to appoint a staffing committee. Councillors with responsibility for employees must ensure that employees are treated reasonably.
If necessary, the staffing committee or the council can authorise the clerk to filter incoming emails and block a particular councillor if the problem cannot be resolved. The troublesome councillor can be asked to communicate with the clerk only through the chairman or another councillor. It must be made clear to the councillor at fault that their behaviour is unacceptable to the council; the clerk cannot be left to cope with the situation alone.
In a nutshell, although it may wish to act discreetly, the council must somehow take action to protect the clerk. In the case I've recently experienced, the chairman and the staffing committee made it quite clear (both in meetings and informally) that they fully supported the clerk and disapproved of the inappropriate councillor behaviour. In that case, the councillor eventually resigned.