Last Christmas my Dad gave me one of his old guitars - a Fender Strat. He is right handed and I am left handed, so he flipped the strings over so I could play it normally.

Though he and I have both had a go at getting the intonation right, the standard bridge doesn't have enough movement in each of the saddles to intonate the strings correctly.

I was wondering, would a left-handed bridge like one of the ones here help to improve this? Or is the problem elsewhere? Any advice would be appreciated.
Could be the way your strings are seated in the nut or the action on the saddles. There isn't any difference in saddles on a standard bridge that would prevent you from intonating.
Poster above touched on it. Id look at the way the strings are seated in the nut.

As you had to flip the strings over, did you also check out the nut slots? Remember the high E nut slot is way thinner than the low E string diameter for a good fit, plus each nut slot in turn in between.

The low E nut slot is far too large for a high E string.