Tithing, from my understanding, only funds churches, temples and any church properties.
... which were all (except temples, perhaps) used to materially support the Prop 8 campaign.
Not really. No tithing funds or any other funds contributed by the Church membership to the Church itself were used in the campaign. The Church asked its members to contribute directly to the campaign itself (which I believe was wrong), but our tithing funds are used specifically for building and maintaining our facilities, printing scriptures and educational materials, funding our universities, etc.
What did the Church use to ask its members to participate in the campaign? Wasn't it the LDS Church's facilities?
For instance, when
this letter was typed, was it done on a computer that was paid for by tithes? It mentions that it was to be read in sacrament meetings; where are these meetings held? Do tithes pay for the utilities and upkeep of this space? How were the internet connections paid for that allowed this letter to get from Salt Lake City to wards in California?
Marketing isn't free. Neither are volunteer recruitment campaigns. It seems clear to me that the LDS Church did both, using the resources and infrastructure that was funded largely with tithes.
I think that most members of any organized religion have issues with their Church's leadership's stance on issues from time to time, but from a logical perspective, we have to weigh the pros and cons of donating our funds to our Church. If all members of the LDS Church were to stop contributing financially to the Church because they didn't like its stance on Prop 8, there would be so many truly beneficial Church programs that would come to a grinding halt that the effects would be felt worldwide. Our humanitarian fund, for example, feeds and clothes hundreds of thousands of people following natural disasters. Our education fund helps truly impoverished young men and women get educations so that they can contribute to their own communities in third-world countries. I don't want to see all of those things taken away.
Wait - I thought you said that tithes only pay for facilities, educational materials, universities, etc. No? Are these other programs directly linked to tithes, or are they directed funds that you give specific donations to, knowing that the money you give will be used for that specific purpose and nothing else? If so, that's not really what I'm talking about: I'm not saying that because the Church is "bad" or something like that, it's wrong to give them any of your money; I'm saying that the entire outcome of donating has to be considered when evaluating the morality of the donation. If you're giving money that you know will specifically go to, say, humanitarian aid, then the fact that it passed through the hands of the LDS Church doesn't necessarily mean that this decision is bad. My issue is only for cases where the money actually funds the anti-same-sex marriage campaign or (IMO) similarily immoral acts.
Also, I touched on this issue before when I acknowledged that individual Mormons may feel that the good that the LDS Church does outweighs the bad on this issue, but the Church's support of Prop 8 and its other anti-same-sex marriage initiatives does work into that balance between good and bad. And the support of the Church in this regard is something that, IMO, all tithe-paying Mormons bear some measure of responsibility for.
However, it's not an either-or scenario like you're making it out to be. Not giving money to the LDS Church doesn't necessarily mean that people in need will suffer, because this frees you to donate more to other organizations that do similar work. Will the world be worse off if you give your donation to, say, Doctors Without Borders instead of the LDS humanitarian fund? Probably not, IMO.
This might help to illustrate where I'm coming from on this issue: I've mentioned before that my wife is Catholic. I find myself in situations fairly often where I have the opportunity to donate to various Catholic causes, which I used to do from time to time. However, when I realized the harmful stance that the Catholic Church was making on the issue of same-sex marriage (mainly when the issue was before the courts and the legislature here, around 2004-2005), I stopped all support. I now refuse to give any money to any Catholic organization. I do my best to make good charitable donations through other (mainly secular) organizations, but I boycott the Catholic Church.
I realize that this is easier for me as a non-member than it would be for someone who actually belongs to the church they're boycotting (although my position has caused some friction at home and with Catholic relatives), but hopefully this helps you get some sense of my approach to this issue.