Once the baby is born you can do a DNA test very easily just by swabbing inside the mouths (mother included) with Q-Tips. There is a company in Oregon that encourages you to use your own Q-Tips so that way you don't need to wait for the test kit to arrive. It costs about $150. The courier cost is about $70, if you have the patience you can use the regular post for $5. This test will show paternity (or not) but will not be admissible in court. But if it shows you are not the father, she is unlikely to press the matter. If you need something admissible in court, respectable clinics will do and certify the swabbing if you bring in IDs and pay them $20 per testee.
If the mother refuses to cooperate in the test, that would not look good, either to you or to a court.
If the baby is yours, the court can order maintenance. Hopefully she will go to court in the province where the payments will be set at a much lower level. If you move to another country it is unlikely she would be able to enforce any court order.