In nearly every (rational/at least remotely rational) conversation I have had with a Mormon (and a handful of people from other religions as well, but most of the conversations are with Mormons) about religion, I have heard the arguments "But Mormons are nice people" and "The church makes me better than I would otherwise be". This is not a rebuttal. It is not a counter-argument.
It may well be the case that every Mormon, every Christian, every religious person on the planet is a good person. It's not the case, but assume that it is. It is not logical to say that the religion practiced by a person is true because the person practicing it is "good". There is no logical cohesion to that argument. It is not sound. A person may be a good, contributing member of society and also believe that we have never landed on the moon. This is a false belief, and sufficient evidence has been presented to contradict anyone who believes it. People have the choice to refute this evidence or to deny its existence or validity. But that doesn't mean that they are right. And the fact that they donate money to starving children in Africa also does not mean that they are right.
The LDS church and many other churches engage in humanitarian efforts. They engage in disaster relief efforts. They engage in community service in some beneficial and key areas. I have not disputed this at any point in time, and anyone who is aware of the facts would not either. In fact, I have in the past (on at least one occasion) linked to a statement by the LDS church declaring that they donated roughly $1 billion in money and in kind to humanitarian efforts over a 25-year period starting (I believe) somewhere in the early-to-mid 80s. That is significant.
The fallacy is that the church is true because it engages in these activities. That is illogical. It is a non-sequitur. So, how about the argument that "the church makes me a better person, so it is good for me to belong to it". Bertrand Russell had an excellent response to this argument. If something is true, then you should believe it. If it is false, then you should not believe it. If it cannot be proven whether it is true or false, then you should withhold judgement on the matter.
To me, the biggest fallacy here is that Mormons insist that their religion is all-or-none. In fact, countless talks given in the church's semi-annual General Conferences state this very thing. That it is either all true or it is all a bunch of lies. This black and white thinking is what leads to this logical fallacy. It is the fallacy of insisting that one believe that everything about the church is good or that it must all be bad. If any one part of it is questioned, all of it must be rejected. All throughout history, we can see examples of people and organizations (ideological, religious, political, etc) where some of the things they do are good and other things are bad. Hitler killed millions of people during his rule as Führer, but he also built miles upon miles of the Autobahn and many other things that were very good for Germany. Likewise, Stalin industrialized Russia. To use the same argument on them would be utterly nonsensical. To claim that everything Hitler or Stalin ever did was evil is clearly false. It is easily refuted. To say that one must accept all of the teachings of Mein Kampf or none of them is equally fallacious.
So, to use this argument of "the church does good things, makes me a better person, therefore it is good/true" is fallacious. To say that "I have benefitted from the church's home teaching program" is fine. That's great. So, programs like the home teaching program the LDS church runs should be adopted. Let's all care about the people that we live near. Let's visit them, see how they're doing, and help them if they need anything. But let's not say "therefore, Noah built an ark that fit all the animals". That's not logical. And refuting claims that the doctrine is false by citing policy is illogical.
Also, I submit that it is illogical even for you to claim that your religious beliefs or your church make you a better person than you would otherwise be. Surely, the belief that one should be empathetic to the feelings of others is one that can influence a person to be more compassionate. I doubt many people would argue that a higher level of compassion would make a person more immoral. However, the fallacy lies in the claim that this belief in being compassionate to others is in some way related to the existence of an all-powerful being or in a blissful afterlife where everyone will live peacefully and joyfully for eternity. If belief in a god--any god, or one particular god--were necessary to show compassion, it would be an easy thing to prove that atheists are not compassionate. But this isn't the case. In fact, historically (and presently) atheists have been among the most compassionate people in existence. The majority of atheists opposed slavery when that was a question. The majority of atheists supported equality for blacks and women. The majority of atheists support equality for LGBT people. If anything, there is a negative correlation between one's belief in god and one's compassion for others.
So, my suggestion then is that it is practical beliefs--such as having empathy for others and acting according to that empathy, respecting the rights of others, and so forth--and not religious beliefs--such as the existence of a god, the evils of masturbation, the power of prayer--that make a person good or bad. It is possible to separate belief in angels from the idea that one should not steal another person's property. It is possible to believe that murder is wrong without believing that there is a devil who will torment you forever if you do it. And I say that if you truly want to be a better person, hold on to all of the beliefs that actually make you a better person and cast aside those that do not. Cast aside any beliefs that are false. Cast aside any that harm other people (such as the belief that homosexuals are evil or that people of other faiths will go to hell when they die).