Have him killed does not mean murder him in this context which should be pretty obvious since that's not what they actually do. It means put to trial, and the gospel is saying that they did not intend for it to be a fair trial. The reason that it would cause problems specifically on the feast day is because that would not have been allowed according to the law.
I think that we can, for a start, establish that no trial could take place on the Sabbath, no?
Finding sources for it not being allowed specifically on the Passover by searching on google is hard since it's very hard to get any relevant results at all. If you want to examine the claim further you can look at that book and it's sources.