It's actually not hard to do, really. But you will need to talk to the parks director, or find out if anyone is in charge of maintaining that particular park. And also if there is a local club who designed it or course designer - just common courtesy. I would suggest two options:
1) consider leaving the pins where they are and just changing the tee locations. This will make life easier, and when you propose the idea to the parks people, remind them this won't cost them a dime. That's probably their biggest concern. But you need to have a map of the current layout and a map of your propsed layout to show them. Point out the lack of safety concerns in your new design and point out the increased challenge as well.
2) moving baskets is a pain, but we've done it in the quad cities. We did it by hand (shovel, actually, ha ha). Dig on one side of the basket down to the bottom of the concrete, so you've got a slope. Chain the basket to a pickup truck or skid steer (easy to rent), and drag it out of the hole. We left the chains and cage on the basket, just to save time and didn't end up with any bending (but be careful, just in case). Have 6-8 guys throw the basket into the bed of the truck or bucket of skid steer, and move it to the new location. If you can't get an auger to drill holes, you're back to digging by hand (which is a pain). It's doable, but not "fun". We left the original conrete on the pole and just buried it again. Some people like to break off the concrete and repour, but that's kind of a waste, really.
Either way, you need to have a new layout proposed to the parks people and have their blessing (and help if possible). they may be able to throw a parks guy your direction to make it easier. Sometimes if they're union, they'll want to do all the work themselves with you supervising - that's really nice, but don't count on it...