With ~540 Watts (includes transformer losses 0.9 PF) the in primary current will be 2.5A the feeder will operate at temperature only slightly above ambient. Adjusting the wire resistance for 45 degree operating temperature will produce results as follows:
For # 12 CU or # 10 AL ~ 3.4 % voltage drop
For or # 10 CU or # 8 AL ~ 2.15 % voltage drop
Since primary breaker should be no more than 15A,2p both of the above are possible, however, ground fault at transformer location will produce only 52A (assuming you provide #6 bonding conductor) that would delay breaker operation beyond 5 seconds. The site and the 3 kVA transformer local ground may improve the time, if really good – but typically the path via both grounds is over 10 Ohm. And since livestock is particularly sensitive to even small step voltage (don’t have shoes :=) we do not want to have currents flowing in earth anyway.
2 # 10 CU + #6 CU bonding conductor will provide 77 A cct. Which should clear fault under 5 seconds.
#10 AWG CU - resistance = 1.12 Ohm/1000ft for conductor temperature of 45°C
#6 AWG CU - resistance = 0.443 Ohm/1000ft for conductor temperature of 45°C
A single phase to ground fault 120 V / (1.12 + 0.443) Ohm =~ 77 A (98% of all faults are phase to ground)
You may find precise data on wire resistance at particular operating temperature using the Voltage drop calculator found at
https://www.mc-group.ca/voltage_drop_calculator.htm It is free for the data you may need.
I am not sure what the bonding requirements in your jurisdiction are but #6 CU is pretty common.
If you do a lot’s of work on farms you may find this link useful.
https://esasafe.com/assets/files/esasafe/pdf/Electrical_Safety_Products/Bulletins/10-23-3.pdf