As a currently employed door supervisor in Scotland I will give you some advice on this from my own experience and research.
First off, legally yes you can carry handcuffs, using them is a whole other ball game. You should have attended a training course in their use plus yearly refresher training, if you use them you have to ensure that it is proportionate, reasonable and justifiable. What is your endgame with this member of the public? To handcuff him, walk him off the site (if he even entered) and release him? Handcuff him just for looking into the site as a way of telling him to stay away or to handcuff him to a pole so you can keep an eye on him all night? All of those would be a one way ticket to an assault charge at the minimum.
If he assaulted you and you used reasonable force to restrain him while waiting on the police to ensure your own safety, in theory you could use handcuffs but it wouldn't necessarily be justifiable in every situation. If you were rolling about on the floor with him in a rural area where the police could be a while and he was exceptionally violent then absolutely it would be justifiable to use mechanical restraints as it is safer for both you and the person being restrained.
As well as training and their appropriate use, you also have to make sure that your company and the client agree to you carrying them, if they say no and you do it anyway it's not a legal issue but it would be a sackable offence. If they say yes then fire in, however that brings me on to this point.
Insurance should also be very thoroughly checked, if your public liability insurance doesn't cover the use of handcuffs then should you apply them in the wrong situation, apply them too tightly or even apply pain using speedcuffs and the person is then found not guilty you can bet they'll attempt to take civil action, now while the court couldn't prove them guilty beyond reasonable doubt, the civil suit will be far less lenient and is based on the balance of probability so you could end up in a pretty bad place.
Finally the type of cuffs you use is also an issue. Speedcuffs are designed to be used in pain compliance, ultimately that's against the physical intervention training set out by the SIA (as terrible as the training is) and as it's used to cause pain some may say it's an offensive weapon, I will leave that their as I'm sure that'll cause a whole other discussion!
Overall while you may wish to carry them, I'd personally say don't bother, having done sites, events, pubs, clubs, and corporate I can only think of 2 occasions in 4 years when using them would have made a huge difference to a situation and that's really not worth it.