The OP didn't make it very clear whether or not the land they refer to is a public footpath or open access land or both. But either way, although dogs are not specifically mentioned in the Acts of Parliament, it is generally considered that they are 'usual accompaniments' to walkers and are therefore 'permissible' on footpaths, other Rights of Way and open access land, provided they are 'under control' and in the case of Rights of Way do not deviate from them. However, as far as I am aware the farmer is under no 'obligation' to provide access for dogs, prams etc, only for pedestrians. (There are of course a few exceptions when dogs can be abnned from farm land, for example for up to six weeks when lambing).
As already mentioned, if you only talked to the farmer about this at the weekend, don't expect anything to change in the next week or so. Be reasonable! And as b r said, why not offer to pay for the means of access if you are the one who needs it? I would comment, at least you talked to the farmer first; don't get the Access Officer at your Council involved unless you really feel you have to.
If the route is a footpath or indeed open access land, do you keep to your side of the agreement with regard to keeping your dog on a lease (Right of Way i.e. they should be kept within 1m of the line of a footpath or 2m from a bridleway or byeway) or within your control (open access)? Do you clean up after your dog (legal requirement on some footpaths and considered good practise and polite)?
In the case of your in-laws, I would question the Access Officer's reasoning behind the request for alternative access to the cattle grid… Is it for disability (i.e. guide or hearing dogs), prams, wheelchairs or some other reason? I would say it seems a bit odd, unless there is proof of frequent use by a person with a guide dog or similar. But if they don't like the route of the access, they can ask for the line to moved away from their main drive / away from their house, particularly if they can show it is intrusive, people don't keep to the line of the footpath, don't clear up mess after their dogs etc. Their solicitor should have made them aware of the presence of the FP and their legal obligation to maintain it and its access at their own cost when they bought the property.