Fowgill is correct, the certificate is only needed for commercial use.
Welfare rules still apply, but not the certificates etc for private use.
One Interesting point re loading of trailers, this was a confusing area with people thinking they couldn't tow a trailer with a theoretical max capacity that would be above their vehicles limit even tho they never loaded it that much (and police were trying to say that too) , so people were downplating trailers to match the vehicle towing. VOSA has now clarified the position a while back. If you passed your test pre 1997 there is NO problem towing a trailer with a max weight capacity above that of the vehicle towing it, and it does not need downplating. All that matters is (as common sense and reason would dictate) is that your licence allows you to tow the actual weight you are towing (post 97 licences would need a trailer test for this, pre 97 is fine up to 3.5t). And that the weight of the trailer and loaded horse etc doesnt exceed the train weight of the vehicle (which is the point ellied was addressing).
Really if you are towing with anything less than a Landy or similarly giant monster, you are likely only to be able to legally tow a trailer with a horse in if it is a single horse trailer eg Ifor 403 and towed by a certain make and model, eg a Nissan Xtrail could tow a 403 with a single horse in it, but many other SUVs cannot. Double horse trailer will certainly need a Landy, shogun etc for towing it loaded. Post 97 licences will need a trailer test too.
To give you confidence about the rules see the excellent official guide for horsebox and trailer owners published by VOSAhttp://www.dft.gov.uk/vosa/repository/Horsebox%20Guide%20low%20res.pdf