Although I don't know the exact figure off the top of my head, I'm pretty sure many credit cards will usually flag purchases of +$1k for fraud unless the customer confirms in advance, but in addition with merchant fees on your end, 3% of $5k is a lot in merchant fees.
For a project in that price range, usually it's better to use an Escrow account because it provides piece of mind to both you and the client (it's always easier to put money in a mutual account vs. paying all up front and hoping the contractor doesn't go rogue) and logistically the volume you're talking about make it ideal in place of PayPal (which is fine for light escrow - as in under $500 - but PayPal can be hell for business owners in larger disputes).
You might want to have a look at Escrow.com and see if it's a good fit. Another option you might want to consider is whether the bank you have your business accounts at has an escrow division and if so, ask whether you might be able to get a discount on the service.
I'm actually in the middle of setting up an escrow account myself so I can't realy provide figures, but I think usually 10% is the transaction fee across the board regardless of vendor. While it is a bit higher than merchant fees, the piece of mind provided by Escrow accounts is probably worth the premium since when using a credit card billing can be more difficult.