From what I understand the Torah only forbids a man from sleeping with his wife while she is on her period, until she goes to the Mikvah. The Rabbis decided to add the additional days as a stringency to make sure she is pure and the period is over.
I am not a very observant Jew but I would like to observe the family laws of Nidda, but to not be with my wife for two weeks out of the month is too much.
Are there any modern opinions or movements that allows the woman to go to the Mikvah at the end of her period and not wait a whole week?
There are no opinions which allow going to the mikveh without the seven clean day counting. In fact these seven clean days are of Biblical origin, however depending on the nature of the bleeding, sometimes Biblically she may go after 7 days from the onset of her period and sometimes Biblically seven clean days are required. The Rabbis instituted to always keep the seven clean days regardless of the type of bleeding so as to avoid confusion that may lead to a most severe transgression of having relations with a Nidda. So in fact without the seven clean days there is a doubt she may be violating the Biblical prohibition.
That being said, your frustration is understandable and it takes time to get used to the niddah system. Ultimately experience shows it pays off greatly, and will increase your marital harmony physically and emotionally.
If at this point it seems impossible to you consider the following. If you are attempting to become pregnant this will give [usually] 9 months without her being a niddah. If you are not attempting to get pregnant, your wife can speak with her doctor, there are many contraceptive pills which you could arrange so that she only gets her period once in a 2 or even three months, and sometimes even longer. You can use this method for now to alleviate this problem.