My experience is people either want something around 1" - 1.5" (I like 1.5 bc it's big enough for a good design without being over the top) or outlandishly big. I haven't sold the big ones myself, but someone I spent a lot of time sharing a table with invested in some experimental 3.5" badges and they sold remarkably well, and the big of extra size adds a bit of extra profit margin.
(Idk how much experience you have with this sorta stuff so please forgive me if this next bit comes off as patronizing.) If you're not planning on selling the, or not doing it often, I'm not sure it's worth investing in a button maker personally when there's plenty on online places that will make you buttons cheaply and at a pretty good quality and a reasonably short run. I wouldn't pay more than £1 for most button badges around the size you're talking, so that's (quick currency conversion later) 250ish badges just to pay for your cheapest maker, plus labour, vs sending it to site (I use Zapcreatives) for 100 badges (often with bonus badges so closer to 120-130 for the price of 100) at £25. So unless you're hugely dedicated to the investment and want to do it all yourself, I'd lean towards getting it done by an online site, or at least while first dipping your toes into it.