There are only two ways to increase water pressure. 1. through elevation 2. booster pump. Increasing the line size out to the barn will only minimize your pressure loss at a given gpm.

I'm guessing that you want to increase the pressure at the spigot over what a hose would give you, not over what you get at the house. Pumps are just future trouble.

Pex is smaller than copper, so 1" pex is closer to 3/4 inch copper than to 1" copper, but you mitigate the pressure drop by having fewer fittings and thus freer flow.

You can get 200 feet of 1.5 inch pex for roughly $600 and that would give you a quarter of the pressure drop that 3/4" pex would, half the drop that 1" would eat up, and about a sixth of the drop over a 5/8" hose, roughly speaking. .

Here is a link that might be useful: a random pex supplier