Hi. Rain or snow does not have to have contact with sensor. .... so just sensor under cover will not help
I would forget trying to get a pir sensor to work at that distance.
I would look at GJD wireless pir heads. Very reliable and last 4 years on one 9v standard battery
Mount pir at your gate ..... set detection patern..... also control on and off and schedule from recorder
http://www.gjd.co.uk/products/wireless-detectors