How much of republicanism can we get without making any essential reference to domination? Maybe quite a lot. 

First, whatever job the government has, if carrying out this job requires substantial power that is liable to abuse, then part of your political ideal should be republican democracy (and you should be very leery of even putatively ideally benevolent dictators). In particular, you might want a high level of assurance, born in part through moral recognition, that the government or government officials will not behave like the criminals that they are supposed to protect us against (supposing that this, at least, is part of the job that governments have). Just say no to a charter city run by Elon Musk or whatever.

Second, perhaps you think that one of the jobs that the state has is pre-emptively protecting against rights-violations by way of regulating the accumulation of too much private power (or imbalances of private power). With regard to the relevant types of abuse or rights-violation, we don’t want the government to always wait around until we are actually abused or violated (or are about to be) to take action.

These two elements are both compatible with a political vision that is fundamentally libertarian (if all of the politically-relevant rights-violations are cases of interference with negative liberty, if it is permissible for the state to do the job of preventing or discouraging such violations to be achieved through coercion, if there is substantial danger that without institutional protections such coercive power will be abused). In which case, though I’ve integrated versions of what are arguably the two most important elements in the republican ideal, my political ideal is fundamentally one of negative liberty.

(One might argue that the second element, properly spelled-out, comes to the prevention of domination – and comes to non-domination on Pettit’s account. But I don’t think domination is the power to interfere in this or that way (“arbitrarily” as Pettit would have it). As importantly, more immediate versus more pre-emptive coercive regulation or control of rights-violations (or of other bad political outcomes) are each important for the same reasons and so should be considered part of one political ideal of assuring that these or those rights not be violated by private parties (or these or those politically bad things not occur). In any case, whatever the merits, I imagine this is the move that the defender of republican freedom as non-domination would make in response to what I say here.)