Since you're talking about church, I take it religious beliefs should be on topic here.
There are two commandments that supersede (and contain) all the others. One of them is
You shall love your neighbor as yourself.
If you have not put yourself in the shoes of the other parents, their children, and the nursery staff, that is where you should start. Would you have your own child repeatedly pushed, bitten, have their hair pulled, toys thrown at, etc.? If the answer is "Yes", then you're good to go (continue feeling unduly put upon/excluded.)
He was kicked out for... trying to bite other kids and babies (I honestly doubt this part happens very often as we don't see this much at home)
I don't know how to say this delicately, but you're assuming bad behavior on the part of the nursery staff (misrepresentation/lying?) because you don't see it at home. On the other hand, you don't have a large bunch of babies and toddlers at home every week to see this interaction for yourself. Unless you can see into someone's heart, it's better to assume a fellow congregant is telling "their" truth until you have evidence to the contrary than to do the opposite. It's also a lot less embarrassing when you find out you were wrong.
Would you enjoy volunteering to be responsible for 10-15 kids when one of them pushes over babies in carriers, bites, pulls air, etc., taking up most of your attention Sunday after Sunday? If the answer is "Yes", then, again, you're good to go.
If the answer to either question is "No!", then you need to frame the entire question positively:
How do we solve this problem?
It appears that your child is not ready for relatively unsupervised play with others. That is true of many toddlers of that age, and one-on-one play dates are a good place to start teaching rules of acceptable play (or more precisely parallel play) interactions, not necessarily sharing yet, but the idea of trading when conflict comes up, the no-hitting/pushing/other. It's possible that time outs aren't going to be terribly effective yet, but I would start them anyway. Removing the child from the situation of inappropriate behavior/hurting someone else if gentle corrections are ineffective is completely appropriate (as is getting the parent to come take their child from the nursery if the staff cannot control them). Toddlers react to frustration with anger or crying, since they don't have words to express themselves yet, power to change the situation, or enough impulse control. An emotional vocabulary will help with disappointments as the child gets older, and time outs help with better decision making. Power comes with age and good decision skills.
When the child can play one-on-one without expressing frustration in the form of hitting, etc., then you can return them to group play to see how they do, supervising the first few times.
For teaching time outs, I would like to recommend the book 1-2-3 Magic by Thomas W. Phelan. Read from cover to cover before instituting, please.
About church attendance, it's usually members of the congregation who staff the nursery. They are usually volunteers, in short supply, and also miss the sermon, so they are not asking anything of you that they are not giving themselves. Many parents of the nursery age kids take turns volunteering in the nursery. Doing so might also help provide some perspective on how difficult it is when one child repeatedly makes it difficult for everyone.
So, in dealing with your child, I'd recommend teaching "manners" during one-on-one play, and then moving to larger groups of kids. In dealing with the church, love your neighbor, asking no more of them than you are willing to do yourself.
If you cannot abide by the idea of volunteering in the nursery, then ask the pastoral staff for guidance in helping to solve the nursery staffing problem.
Edited to address new information:
We have a five year old who does like to "play rough" with him, but we have tried to calm that down because we fear that may be part of what's causing the problem.
Although jealousy between siblings is featured in works as old as the Bible (Joseph and his brothers; Esau and Jacob), the actual study of "normal" sibling relationships themselves is relatively scarce, so I will only present behaviors I have seen myself in my family, extended family and my practice.
Many children (especially preschoolers) are jealous of their younger siblings, and since they (usually) can't just march up to them and punch them, they mask their aggression in "rough play".
I suspect this does have quite a bit to do with the toddler's behavior, so you need to work on changing this dynamic at home as well. For this, I would also recommend 1,2,3 Magic as a starting point.
None of this means you have bad kids; it honestly sounds fairly normal for the situation. It's a lot to deal with. You have my sincere sympathy.
An example: my first constantly "accidentally" bumped into the second, knocking them to the floor. In preschool, the first went from angelic to fierce biter immediately after the birth of the second. The second, in turn, was jealous of the first, at seven months crawling over to bite the foot of the first when they were seated on my lap.