Well, this really depends on your ability. If you are more inclined to American accent and possess the ability to acquire it, then no one should stop you. On the other hand, if you prefer British Accent/English Accent and are capable of teaching yourself this, you may do so. However, if you are interested in neither of them, yet you want to sound less Japanese, go for neutral accent.
Neutral accent or reduced accent is an approach to modifying your native accent through a series of learning steps such as changing the way you move your tongue, jaw and mouth to form vowel and consonant sounds. This is the accent that is prevalent among call center agents, which proves to be very useful since they can speak to and be understood by Native English Speakers and Non-Natives alike.
Now, the first step to improving your accent; either you want to have American, British or neutral Accent, is identifying your problem areas in terms of vowel and consonant areas, to name few. This often involves a lot of movements of the jaws, mouth, lips and tongue. Unfortunately, these movements happen to be one of the many challenges for all Japanese learners. Insisting on doing it your way like speaking with your teeth clenched and your jaws hardly moving will get you nowhere. As the saying goes, no pain no gain. So, learn English Pronunciation the right way.
Thank you for reading.