Using Table 8 in chapter 9 of the NEC along with Ohm's Law (the max drop generally permitted by good practice on a branch circuit is 5% or 6V on a 120V circuit) gets you a minimum wire size of 6AWG in copper or 4AWG in aluminum.
Considering that 6/2 UF is $3.00 a foot at the blue borg and 14/3 UF is $0.87 a foot at the blue borg, the transformer idea looks like a good option, until you realize how much 3.6kVA of transformer will cost you -- almost $600 via McMaster-Carr, which is the cheapest supplier I could find, with an alternative part available for about $650 via Platt Electric Supply.
Even with 240V being tapped at the house (source) end of this instead of 120V (so you only need one transformer) -- you'd still be out a good $850, which is comparable to simply burying 6/2 UF. A further drawback created by the transformer idea is the extra labor involved with creating a separately derived system in the barn (the equipment grounding arrangement gets...funky, see NEC 250.30 for details), so simply burying 6/2 UF is going to be your best option.
As to terminating it in the panel? Don't worry about it! The main lugs on a panelboard will easily accept 6AWG! If you're not using a panelboard (as this is a single branch circuit feeding an outbuilding), simply use appropriate wirenuts to pigtail the 6AWG UF to a length of 10AWG for the indoor branch circuit in the barn.