I am unsure what your objection is to the deputy conducting the investigation. Was he/she previously involved or something?
Your question seems mainly centred around a) whether or not it is reasonable for your employer to demand that you apologise and b) whether determining the level of sanction applied with an apology as a criteria is reasonable.
In respect of a), yes of course it is reasonable for your employer to require you to apologise. Ideally of course, they wouldn't need to require it as you would do it anyway.
In respect of b), this is a recommendation from the investigator, but I think it would be reasonable of them to take into account whether or not you had apologised when deciding what level of warning to apply, yes.
However, if the level of warning has been determined (or conditionally determined) before the disciplinary hearing has been conducted and you have had a chance to put across your side of the story, that's where the procedure is unfair. Taking an apology (or lack of) into account isn't unreasonable, it's the deciding in advance that is a problem, if that is what has happened.
So you have several options. You could apologise, go through the hearing, be contrite and acknowledge your (undoubted) error of judgment. You will be given a written warning by the sounds of things. You could then decide whether you want to appeal it based on the procedural error of deciding the level of warning in advance, or you could accept the warning as being a reasonable outcome anyway. If you appeal it based on that procedural error, that doesn't mean the level of warning would be changed.
You could not apologise, because you don't think it warrants an apology or because you are being bloody-minded and don't want to apologise if they are demanding it. You will then go through the disciplinary hearing, they will take into account the fact that you have refused to apologise when deciding what level of sanction to apply. You could then appeal that if you want to, probably on the same procedural error of deciding in advance what level of warning to apply.