Solution:

Antonyms or contraries are the words that have a meaning that's different from the original meaning of the word. 
thus, going according to the hint we will dissect each and every option and find out the most suitable option. 
Starting with option A, the word is lowered. This word means to reduce. This is analogous in meaning to the word bated as bandied in the hint. thus, the word will be a reverse with bated and not the antipode. 
Option B is increased. This word means to enlarge or rise. This is the contrary in meaning to the word bated. According to the hint mentioned over, this option is most suitable. thus, option B that's increased can be the correct option. 
The word lessened means to reduce. This is a word that has a meaning analogous to the meaning of the word bated. thus, this option doesn't fulfill the conditions of the question. Hence, option C is incorrect. 
Option D is Lapsed. This is also a word that has a meaning analogous to the meaning of the word bated. The word lapsed means to reduce or retardation. thus, this word is the reverse of the word bated but not the antipode. Hence, option D is incorrect. 
therefore, option B that's increased is the correct answer.