Learn English – At its own discretion

1. Yes, “has been having” is perfectly fine in English.

2. In your example sentence, “has been having” does not work.

“Has been having” is the continuous aspect of the present perfect tense “has had”. The continuous aspect requires either an action that has been ongoing or an action or state that has repeatedly occurred in the period given.

In your sentence, the verb used (‘to have’) does not indicate an action, but a state. That rules out the first option (an action that has been ongoing). “Has been having” would then only be right if the state has occurred repeatedly (i.e., on several different instances), which is not the case here: your company hasn’t opened, and then closed, a whole series of bank accounts with the bank in question. Therefore, the continuous aspect is not a possibility, and the simple present perfect must be used instead:

My company, XX, has had an account in your bank for the past three years

(Note also: an account for the past three years)

If, on the other hand, you are talking about something that really is a repeating occurrence, you can use the other construction; compare:

I have been having headaches for a month now.
I have had a headache for a month now.

In the first sentence, I have been getting headaches on and off (but repeatedly) for the past month: they come and go. In the latter, I have been in a state of having the same, permanent headache for an entire month (ouch!).

As a Brit, I find at natural here. This is the NGram for BrE corpus...

But apparently, Americans are just as likely to use in. Here's the AmE NGram...

Of course, there's no real concept of "right" and "wrong" here beyond what others normally say. If you're writing it yourself, and your target audience is British, you should probably go for at (but given the US is the world’s most litigious society, it would be unwise to argue with an American lawyer who prefers in! :)