What kind of closet will this be? Can't put a panel in a clothing closet.
A bathroom needs a 20A because a hair dryer is 13A all by itself nevermind the curling iron or hair straightener often set out to warm up while it's in use. Also whatever else, and lights, fan, heatlamp if that's on the circuit too.
I get your idea that it's YOUR house and YOU can cheap out on the electrical system and use your smarts not to overload it. However a) don't overestimate your sense of load size. b) I certainly hope for your resale value that you are not the last human who uses this house. And c) I fail to see the value of cheaping out on electrical. Given the astonishingly low cost of the parts, you should be optimizing for performance not economy. Case in point:
The ampacity of the subpanel
First, separate the supply breaker from the subpanel proper. You can have a 30A breaker in the main feed a 225A panel (the 30A certainly won't melt the 225A panel!) The reverse is not true, obviously. You would go with a higher amp subpanel mainly so you have extra breaker spaces for future expansion, because running out of spaces is a nightmare that is so easily averted right now, when you buy the subpanel, and extra spaces are cheap. So in your case I'd go for a 30 space for instance. Finishing with 60% of your spaces unused is a good place to be. It's so easy and cheap to just add circuits if you have the panel space, so things like under-sink water heaters, electric car, welder, become easy decisions.
Beware of double-stuff breakers. Many panels advertise "24-circuit (12-space)" - the 24 circuits relies on double-stuff breakers, which are more expensive. Those are bad news because increasingly you are required to use exotic breakers like GFCI, AFCI, 2-pole and the like, and those cannot be had in double-stuff. So you are stuck with the lesser capacity e.g. 12 circuits. It's cheaper to get the number of spaces you want, so you can use the simplest breakers allowed.
Which panel? The breakers must match the panel. Eaton CH panel, Eaton CH breakers. Most manufacturers make several incompatible lines (CH/BR, Homeline/QO, etc.) The ruling factor is the cost of exotic breakers such as GFCI, AFCI, 2-pole GFCI and the like. So you can cross off several models. Those exotics are being forced down our throat, so get used to them. I am also not a fan of tenant-grade cheapies (Homeline, BR). Siemens has a lot of promise: quality panel with sanely priced exotics.
The panel should be fed with a 2-pole breaker in the main panel, unless you are doing something intentionally weird, but then why would you use /3 feeder? You should know this, by the way.
I'm a little concerned with a knowledge gap here, and encourage you to hit a library for a book on wiring houses, and read it cover to cover. You don't want a knowledge gap. You especially don't want one after the drywall is up but before inspection. If your skill is lacking, do the works in EMT conduit so it's easy to fix later!
The circuits
Now, you say you want to run /3 cable. Either you don't know what you're saying, or you intend to double the circuit so you get 2 circuits out of each cable (a multi-wire branch circuit). That's awesome. (MWBCs not unfamiliarity).
I was a little worried that you were underpowering things like the bathroom and kitchen, but if you intend to supply them with a /3 and MWBC, that means double the circuits, and the cook will thank you. However MWBCs absolutely require a 2-pole breaker, which means you will need 2 breaker spaces for each one. Like I say, you go through spaces faster than you expect!
Honestly MWBCs are falling out of style because so many circuits require GFCI or AFCI. And that requires either the uber-exotic 2-pole GFCI/AFCI breakers, or correct branching at the destination and then GFCI or AFCI deadfaces or receptacles. And the savings is pretty small compared to running 2 separate /2 cable runs and avoiding MWBCs altogether.
The bathroom(s) require following one of two strategies:
Each bathroom gets a dedicated 20A circuit for ONLY loads in that bathroom, including wired loads like lights, fan, etc. I am not a fan of this one, because BOOM you are now in pitch black, holding a red hot hair straightener 1/4 inch from your ear, and can't even set it down because you can't see to not set it in the sink full of water. (however if the hardwired loads together exceed 50% of circuit capacity, looking at you fan-heater, that circuit cannot serve any receptacles at all, and this method doesn't work.)
A dedicated 20A circuit serves only receptacle loads in any number of bathrooms. The hardwired loads are served potentially from other circuits that serve other rooms.
There is one circuit that you really will want to be 100% dedicated, and that is the refrigerator. Code requires a GFCI since it's in the kitchen, even though that's not really the use-case GFCIs were intended for. And you really don't want any other appliance tripping the fridge and ruining your food. For this reason GFCIs on fridges are a terrible idea. I say install a dedicated circuit with a single (not the usual duplex) receptacle, so only the fridge can possibly be plugged in. Use a GFCI breaker, and I would certainly never tell you to switch it to a plain breaker 5 minutes after the inspector walks out the door.
Best Answer
Why not bring the 4 AWG all the way into the panel and splice it there. If you are using a 20-A breaker, then you could use 6, 10 or 12 AWG to go to the breaker. The neutral and ground of the 4 AWG cable could be connected without splices if the neutral and ground bars would accept 4 AWG.