You should be able to cut a circle with a trammel arm on the router or make a simple circle cutting jig out of 6mm ply and a point (nail)
cut a hole in the ply for the cutter to fit through and drill for the fixing screws to the router base plate. put a point for the circle centre offset from the router centre allowing for the cutter radius. put the point in the timber to be cut then revolve around to cut the circle. Route a bit deeper with each cut til the depth is reached. You could also rough cut the circle with a saw before finishing with the router.
You can use any size bit within reason as long as the radius from the centre of your point to the edge of the bit (ie where it will leave the cut edge of the circle) is what you want. However, I would use a quarter inch bit - there is less spruce to cut and therefore a generally easier job.
A straight cutter or a spiral up cut bit would work fine.
For a one-off job, I simply screw a bit of 3mm hardboard to my router base and stick a brad through the hardboard into the workpiece. If I absolutely cannot tolerate a hole in the workpiece, I stick the brad through a bit of thin wood and use doublesided tape to stick that to the workpiece. In that case you have to "crank" the temporary router base, making something that looks a little like frying pan where the router sits on the "pan" and the nail goes through the "handle".
There are similar methods to use bandsaw or even a table saw to cut out circles, I can probably find a reference if you need one.
If you have a drill bit the same size as your guide bush that makes an even easier way to "attach" your router to your bit of MDF/ply with its nail pivot point. Also means you don't get the power cord twisted round the machine as you go round. Like Chris, I use a 1/4" bit.
Yes the Jasper jig does this very easily. You do need a pilot hole for the centre pin to locate in. The main reason I got the jig is it can be used to cut small discs ie less than the size of the base of the router. It is repeatable and needs very little setting up.