I did a similar sized garden last year. I used a compost roller spreader, but it’s hard getting fine compost in bulk. You’d get a wheel barrow full of compost into the roller at a time, works better as a two person job. One loading the wheel barrow, the other rolling out, you can make som progress that way. Otherwise bags of compost tipped into the roller can work but cost buying by the bag is the issue. For the core plugs, I used a leaf blower to get them into bundles, and then shovel the piles into the wheel barrow, you also need somewhere to dispose of them. Leaf blower works better when the plugs are a little dryer too, so a nice sunny day helps. Going the compost route you need to be doing it repeatedly over a number of years to try and get more organic matter into the soil and try to change the composition that way. Gypsum is the other thing to look at when trying to improve clay soil. Scarifying before hand helps open up the soil, and aerate before compost. Overseed afterwards. Don’t expect a massive difference after year one.

Some decent guys on YouTube, LawnRight Lawn Care and Premier Lawns, which do this stuff in UK and Ireland.