I would not use 'bakari' in that sentence unless I wanted to sound unusually casual. The word would give the sentence a highly Japanese-like character that the native speaker would not expect to come from someone who has been studying Japanese only for three months.
Bakari with counters refers to a rough extent or approximate amount and is therefore equivalent to "about." It has nothing whatsoever to do with the core grammar involved in indicating a specific amount of time.
If it's necessary to imply a guess is being made "三ヶ月くらい" can be used much more normally.