I'm a hillwalker who also rides, but as a caveat, I've never taken the horse out in the Highlands due to not owning a trailer or lorry. Anyway, you're right -- a lot of the trails are boggy, but there are plenty you could ride on. However, your biggest problem will be gates: ones that a human on foot or on a mountain bike can access, but it will be either a narrow gate, a kissing gate, or a stile, and you won't get a horse through. These are usually next to locked five-bar gates that provide the estate with vehicle access. If you had a horse, I imagine you would have to contact the estate and ask for the gate to be unlocked, but it's sometimes not easy to work out who owns what and who you have to contact.
Horse riders have the same rights, in theory, as hikers and cyclists under the Access Laws, but in practice it doesn't work out that way because while estates have to allow hikers access, they can legally prevent unauthorised vehicles from using their private tracks, so they lock the gates that something car-or quad-bike-sized could fit through, which unfortunately affects things that are horse-sized as well. Not enough people ride these trails to have thrown any kind of meaningful fit about it, so as far as the estates are concerned, they do not need to provide riders with easy access.
These people offer commercial trail rides throughout the Highlands, including (based on their website) multiday treks. You should try to contact them, as they will know how to deal with the access issues on the trails they use.
http://www.highlandsunbridled.co.uk/.