Doubt is the distance between what we expect of God and what we experience in reality. What could be sinful about this?
The act of doubt, that is, the admitting of this gap, this distance, is not sin at all. It is a sort of repentance.
If we're going to doubt, we might as well doubt a God worth doubting. I don't bother expending much energy doubting the existence of Chemosh the Fish-Headed God of the Moabites, or the Mormon God or a god who would have endorsed slavery in the American South, or a deist conception of a God who watches us at a distance. Whether these gods are more plausible than a properly Christian conception of God is irrelevant. None of these God enthrall me. They don't evoke worship from me.
This principle, doubt a God worth doubting, makes doubt so much easier. Would the God I want to believe in hold it against me that he doesn't seem real to me?
I might doubt for terrible, sinful reasons. I might doubt God because he doesn't give me what I want, or because he's not an anarcho-syndicalist and I think I might be, or because of widespread human suffering that might not actually be his fault. Forcing God into a false conception of him is wrong; and it's foolish if I don't even like that false conception. However, the doubt itself isn't the sin, it's a symptom of a sinful attitude towards God. Doubt is the start of how that attitude would get resolution.
When I have an idea of a God I really want to love, one with integrity and mirth and mercy and danger, but I just don't see signs of his work in the world, no matter how hard I look, that's good doubt for me.
I love fights. I love wrestling and jokes and crying and yelling and singing and repentance and screaming. I love a good poop, or when I'm sick, hocking a huge loogie. I haven't vomited in years, but when I get a stomach bug, I wish for a puke-o-rama, to get it all out.
If you're angry at God, you're angry at God. Be angry at God.
If you're angry at God, do it at God, emphasis on the at. The Israelites in the Pentateuch wouldn't get angry at God, their dissatisfaction smoldered in grumbling, and this is sin. The Pharisees in the New Testament behave in precisely the same way; notice how many times in the gospels they grumble among themselves—but they never ask an honest question of Jesus.
If you're going to doubt, I suggest going all the way, doubting as thoroughly as you can. Don't say "I'm struggling to believe in God." If God doesn't seem real to you, just say so! Get used to thinking two things at once. This sounds confusing, but it only takes twice as much thinking as you're used to. Scream "God isn't real!" if that's the way it seems to you. But, think, "If God is real, then he'd certainly know better than to believe me when I say he's not, and if he's not real, there's no God to be troubled at all by my doubts." It's very refreshing.
I'm amused by the particularly angry atheists, who study a lot of anti-apologetics to prove to Christians that God isn't real. I think I'm the only one that I know personally who was wooed by their thinking. They hate being atheists, but they hate the Christian notion of God, too. They become embittered, bile and gall accumulates in their souls. They're very logical people, but they don't see the lack of logic in expending energy in disproving a God they don't even like. It's futile to be disappointed at God you don't like for not existing, that's like Jonah cursing the vine.
(There is a different category of angry atheists that I don't think are goofy. They are angry at the Christians who are worth being angry at. I wish them the best of luck. I also want them to think that I'm cool.)
Doubt because that's what you really feel. Don't doubt to punish God, that's silly, that's like when a child punishes his parents by holding his breath. I know one person who would hold his breath until he would pass out. He is now a black-belt in jujitsu, but that's irrelevant. Wise parents know to let the unruly child hold his breath as long as he likes, but not to budge an inch: otherwise, they're teaching the child that asphyxiation is an effective way to get what he wants. I suppose God, being a good parent, would deal with inauthentic doubt by ignoring it.
You don't even have to know, though, which god you're doubting. You don't have to be consistent in your doubts—the biblical conception of God is gleefully inconsistent, "Jacob have I loved and Esau have I hated." The things that everyone in your church says are the really good things about God can seem diabolical to you. You can take your pick of scriptures that you find dull or backwards; I could help you find good examples. Why would God push you away, if you were to meander near him, but with confusing reservations? I'm sure he confuses himself quite often, and won't blame you for your confusion. Christianity has never taught salvation by not being confused by God.