Definitely build it up. It will be fun, you'll learn some stuff, acquire some tools and the bike will feel more like yours. I'm on my 3rd fixed- the first I built up for myself based on an old Giant road frame with a mixed bag of low end parts. Worked great, I rode it for a few years and then wanted a track frame. Came across a like new complete Bianchi Pista in a thrift store (very weird but true) which I got super cheap. Great bike but not as fun. Didn't have the same connection. Sold it last year and bought a nice lugged Italian frame and build up a much nicer conversion. Hands down the best of the lot. Biggest expense was hand built wheels with Velocity Aeros and Suzue Pro Max hubs. But anything built around Formulas will do you.
Just buy tools as you need them rather than a kit and buy good tools. I have some Park tools I acquired as I built up bikes and they are great and I don't regret dropping cash on them. You buy it to use it once and end up using it a lot more than you'd think. Since the bottom bracket and headset are in, really the only special thing you might need is a lock ring wrench. Everything else you can get Craftsman or whatever your favorite tool brand is. Probably some allen keys (I like the Park tools 4/5/6 mm "Y" wrench, it fits everything on a bike), a 15mm wrench for axle bolts. A socket to fit whatever cranks you buy and not much else. Maybe a chain link tool. You can get threaded stems for cheap on Ebay or at a local store. Same with a seatpost. Cranks, it depends on how high end you want to go. For my first one I just bought whatever was cheap and available and it worked out fine. Become friends with your local bike store and fish through their used parts box (everyone has these).
Do it.