There are two ways that you can get from Battambang to Trat.

Via Pailin
By share taxi (though Pailin) to the boarder at Phsa Promtakes takes around 31/2 hrs. Note that when sharing a taxi pay for 2 seats each for more comfort. Of course you can hire a whole taxi if you want. The road is not sealed (just a hard packed dirt road in fair condition) but the scenery is quite nice. From the boarder to Chanthaburi it is about 2 hrs by taxi and then around 11/2 hrs either by bus or taxi to Trat. Time around 8 hrs or more, depending on connection time, on an unpaved road for half the trip.

Via Poipet
The road to the Poipet boarder crossing is good and by taxi will take around 21/2 hrs. After crossing over you can either go to the taxi stand for a taxi to Chanthabur (or maybe all the way to Trat) or take a Tuk-Tuk to the bus station in Aranyaprathet and take a direct bus to Chanthabur. Around 3+ hrs. Then take the bus or taxi to Trat. Total time about 8-9+ hrs, again depending on connection time, over good roads mostly on the Thai side.

You should also be aware that at land boarder crossings you may be asked to show proof for leaving Thailand within the 15 days.you are allowed to stay without a visa. This is especially true for the Pailin boarder.