I think one of the hardest parts of deconstruction is to not hold arrogance or bitterness towards people that don't agree with you. I've already mentioned that you need to remember that you believed the same thing not long ago. Here are some other thoughts on how to pursue humility, kindness, and relationship even when you are in disagreement.
ACKNOWLEDGE THAT YOU MIGHT BE WRONG.
Maybe not on this particular issue, but certainly somewhere in your belief system. David Platt says, "I'm fairly confident that I've got it right on 90% of my theology. The problem is, I don't know which 10% is wrong."
I love Sir John Templeton's thought:
“The idea that an individual can find God is terribly self-centered. It is like a wave thinking it can find the sea.”
Somewhere in your theology, you are wrong. Don't be so harsh or critical of others when you think they are missing the mark.
DETERMINE IF THE ISSUE IS ESSENTIAL, IMPORTANT, OR PREFERENCE.
Augustine famously wrote:
"In essentials, unity. In non-essentials, freedom. In all things, love."
Ask yourself if the disagreement is over something essential to the faith. There are only a handful of 'essentials,' things that you must believe in order to be considered a Christian. (Must is a really strong word...) You could look to the Apostle's Creed as a ballpark if you want.
If the disagreement is over something that is not essential, is it still important enough to you to break fellowship? For example, you may be of the mindset that you can't be in a church that isn't open and affirming towards LGBT. (Or you can't be a part of a church that IS open and affirming.) This disagreement is important because it is personal. So this issue, while not essential (to being a Christian,) might be important enough to you to cut ties over.
If the disagreement isn't over something that is essential or important enough to end the relationship, perhaps t it is merely a preference. It might be fun to debate, but at the end of the day, you can still be in community with people that you disagree with. In my opinion, most disagreements are simply about preferences, but we make them important or essential.
ASSUME THE BEST INTENTIONS OF THE OTHER PERSON.
I'd encourage both sides in any disagreement to assume the best intentions of the other. Don't demonize the opposition or incorrectly assume moral superiority. (Especially if the issue isn't really about morality to begin with.)
Sticking with the issue of LGBT as an example (because that's not controversial at all...), it is all too common for progressives to label the conservative side as bigoted or homophobic. It is common for the conservative side to accuse the progressive side of not valuing the Bible and wanting to tear it down.
What if your assumption is wrong? If you've already made your mind up about the opposition, you're going to interpret your interaction through the lens of that bias. You're just going to try and 'win' the argument. You aren't going to connect with the other. And you aren't going to learn anything either.
And what if your assumption is right? Jesus plainly told us, "Love your enemies." This might be a great time to practice.
If you're willing to engage in tense conservations and work through nuance without over-generalizing, you might learn you are much closer than you think. You might learn things that reinforce your position. You might learn things that make you reconsider.
"It is impossible to teach a man something that he already thinks he knows." -Epictetus
FOCUS ON A MISSION BIGGER THAN YOUR DIFFERENCES. REALIZE THE STRENGTH OF DIVERSITY, FLEXIBILITY, AND HUMILITY.
When Jesus created his movement, he pulled together people from all different economic, political, and social backgrounds. He got them to get over their differences (it took a while) and focus on what the kingdom of God was doing in their communities. The beauty of what was going on overshadowed their petty disagreements. We often lose the sense of awe and mission and turn in on each other.
Paul copies the model of Jesus by planting churches in a wide variety of cultures all over the Roman Empire. Most of Paul's letters are trying to teach people with different backgrounds how to unite under the Lordship of Jesus. The faith he developed had to be resilient, flexible, adaptable.
Since the Reformation, we've often taken the opposite approach, splintering off over specifics, being rigid over non-essentials and creating systems that are not resilient. We should look to Paul's words in Philippians as a path forward:
If you’ve gotten anything at all out of following Christ, if his love has made any difference in your life, if being in a community of the Spirit means anything to you, if you have a heart, if you care— then do me a favor: Agree with each other, love each other, be deep-spirited friends. Don’t push your way to the front; don’t sweet-talk your way to the top. Put yourself aside, and help others get ahead. Don’t be obsessed with getting your own advantage. Forget yourselves long enough to lend a helping hand. (Philippians 2:1-4, The Message)