You will need a ground wire running back to the house. Since you have upsized the wire beyond bare minimum, a fudge factor gets applied to the required grounding wire size, so the required size is #6Cu or #4Al. If it is aluminum, it must be in an insulated jacket; copper can be bare. The ground rod is not enough.
You always need the ground wire back to main. Because it is an outbuilding, you also need an 8 foot ground rod. In fact 1 ground rod is not enough unless you have done a resistance test and confirmed <25 ohms. If you have not, you need 2 ground rods. None of these are a replacement for a ground wire because a 25-ohm ground cannot return enough current to trip a breaker. (At 120V, 25 ohms won't flow more than 5 amps, which will certainly not trip any breakers.) Dirt is a very poor conductor of electricity, which is why we mine copper :) The reason for ground rods is equipotential bonding and ESD discharge, not fault-current return.
That subpanel is awfully, Very Tiny small. You are going to use up those 8 spaces faster than you can blink, and will end up with a huge surplus of wire capacity you can't use for lack of fitting breakers. On this forum we very forcefully recommend using "huge" (by your standards) panels, because bigger panels are cheap and regrets are expensive. Also because modern AFCI requirements makes double-stuffing breakers no longer possible.
Nothing sucks worse than having to hork-a-dork something, or change a panel, because the last guy installed too small a panel, saving the price of a couple pizzas on a $1000++ project. I sure hope he enjoyed that pizza. Given the amp service (150A fully credited), get a 24 bare minimum. ThreePhaseEel and myself would fit at least a 30, having dealt with full panels too many times.
The subpanel need a master shutoff switch before it. The best/cheapest way to get that is to use a panel that has a main breaker. The size of the main breaker does not matter, since you're only using it as a shutoff switch. But it's a waste if the subpanel's main breaker is smaller than the feeder wire's capacity; and that feeder wire is good for as high as 150A. (135A textbook, then round up to next breaker size).
So you will definitely be replacing that subpanel. If you have any aspirations to put this outbuilding on its own generator, I recommend Siemens. Regardless, a generator here cannot backfeed the house, because it is then impossible to interlock with utility power.
That circuit breaker most likely needs to be AFCI, GFCI or both. This will be true of most of the breakers in the panel. So if your plan was to rely on the "8 space 16 circuit" marketing of the panel, that 2n circuits number is a blatant lie. That relies on double-stuff breakers, which are not made in AFCI/GFCI, so you can't hardly use them anywhere.
Separate ground bar; good choice. Neutral must be isolated from ground in a subpanel. When people think of bonding them in a subpanel, I worry that they failed to run a separate ground wire along with the conductors, but you say you ran a #4Cu which is excellent. #8 would be required for 100A wire, but you fit 150A wire so #6. #4 is more than is required and will assure rapid main-breaker trip should that be necessary.