I have a plot and am site rep on a council run site and we are not allowed chickens. They are permitted on a couple of sites in the area, but you have to apply for permission. This is their wording on the subject if it helps.
Chickens
Livestock may NOT be kept on allotments, EXCEPT xxxx and xxxx Allotments, where the keeping of chickens/ducks is permitted, with written consent of the Council and subject to the following special conditions, A- The numbers of chickens/ducks must be manageable; B-the livestock are well and humanely managed; and C- they do not cause any danger, nuisance, interference, disturbance or annoyance to Tenants or to anyone else including members of the public and the owners or occupiers of the neighbouring and adjoining property or plot holders. The Council may decide that keeping livestock on the site would be prejudicial to health or a nuisance and has discretion to suspend permission to keep chickens/ducks on site.
For what it is worth, my experience as site rep has mostly been positive, but I have had a couple of people of a similar mind set to your tenant. Just telling people what the rules are is not heavy handed and if people are doing things without permission, you have every right to tell them there are consequences if they persist. They are trying to ride rough shod over you and make their own rules and that is just not how it works - either for allotments or life in general.
I would say stick by your guns and tell them if they continue to do so, they risk eviction. You don’t have to actually start any steps to do this. That may be enough to stop them. If not, I would say go ahead and start the process. You are not evicting someone for being flaming difficult, they are doing that to themselves