To answer your last question first, Pelargoniums, aka 'Geraniums' in common parlance, which I think is what you have, are extremely easy to root from cuttings so you can cover your back by producing as many plants as you want before moving the parent.

I think a large Pelargonium would transplant fine but I am not so sure about splitting it if it has only one main stem coming from the ground. If there are multiple stems go ahead and divide. They are very forgiving.

Bear in mind also that looking after plants in pots takes more effort and care than those in the ground, especially in dry climates.

Here is a link that might be useful: Pelargonium cuttings

Thank you very much, Flora! Yes, I should have said Pelargonium. Okay, that's what I feared! I'd love to split it in two but there's really no way. I was planning to grow some from little cuttings anyway, because it's such an unusual one. It's not blooming now (which is why I was thinking of moving it/splitting it) or else I'd post a photo. Thanks again.

Your geranium is, as you understand it, a California native and as such has certain sun-loving attributes.
Before you do anything that might cause the plant harm, why not call up your local full-service nursery that sells the type geranium you have. Ask for specific instructions on how to best divide it and, more to the point, when to do it.
In California, just as it is in northern climes, the sun is losing intensity and this is not the time to think to divide plants. When division is made you look for the plant to grow...and in December--OK, its still November...but catching up to December when the sun is still not intense enough to support new growth.

Wait until plants begin to add new leaves...your cue that it will soon start growing and the time to divide.
Taking cuttings is a fine method to obtain many new plants as long as it doesn't interfere with the mother plant getting nourishment.
Don't try to feed a plant into health--its not growing and cant use the food. Wait until the plant tells you its growing....new leaves---then feed gradually as it can use it.

goren - firstly pelargoniums are native to Southern Africa, not California. Secondly, they are very easy and forgiving plants and I would have no compunction in moving or dividing one even in my climate at this time of the year. In warmer locations the Autumn is a fine time to move plants. The one in question is described as having a 2 inch stem and would suffer no harm at all from having cuttings taken. They will root easily even if they have been lying about for several days unplanted. And finally I would not recommend feeding a Pelargonium, certainly not one in the ground, unless you want to have a lot of leaves and few flowers. Besides, the question was asked 2 weeks ago so it may well already have been done.

You should transplant it now. I've moved large pelargoniums at all times of the year, and I think they suffer less if moved at this time of year. Whenever I have to move a large pelargonium, I normally cut it back until it is a manageable height, then using a shovel, I go all around it and slowly use the shovel as a lever to pull up as much of the root system as possible. All of my pelargoniums, regardless of how large they may be, tend to have a relatively small root ball. They are quite resilient, and although they may look unsightly for a few months, it is best to transplant them now while the weather is cool. Plus, with all the extra pieces you cut, you can start a ton of new plants to share with friends and family :)

The large pelargonium in the center of the photo, to the right of the stone path, was actually a volunteer seedling, probably from seed blown over from a neighbor's plants. I've had to move it several times, and recently divided a small part of it, which you can see planted on the far lower left side of the photo.
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Lovely photo -- and many thanks to everyone for the kind advice! I wish my garden looked like that! :)