Does anyone know how much being married would help or not help in a custody battle?

I think showing that you're in a stable and mutually supportive relationship helps - but I don't know that being married as such makes a difference... It depends where you are.
In Australia, for example, 'the best interests of the child are considered paramount'... That means that the child heads to the warmest most loving home possible, where all its needs will be fulfilled... At least that's what it means technically, I guess... But anyway - what I'm trying to say is that that doesn't have much to do with a wedding certiificate.

I was thinking that maybe it would show more stability for the child? Less chance of the child getting to know their parents significant other and becoming attatched and then have them possibly split up? My fiance' and I are obviously getting married but we were wondering if we should move up the date.

If you're talking a married parent fighting for custody of a child, if the other parent is not married then it could be a factor in that the married parent may be able to present evidence of being able to provide a more stable home. All in all it's a minor consideration but every little bit helps.

I was thinking that maybe it would show more stability for the child? Less chance of the child getting to know their parents significant other and becoming attatched and then have them possibly split up? My fiance' and I are obviously getting married but we were wondering if we should move up the date.

I wouldn't necessarily move up the date just for that reason.