We were just there last week. Lots of people bring picnic food in. There are some grocery type stores very close by to Tivoli (Irma near the British toy store for example) and you could stop there before hand, or duck out, eat elsewhere, and then shoot back in. Also the train station is next store and there is no shortage of dining options there. We ate at the Anderson bakery (yummy!) which is right there. You do have to duck out, but you can go back in. I know that sounds weird, but trust me. There's also a wide range of price options when eating at Tivoli. You can go 4 star to just hot dogs.
Yes there are lines, but they move. I would consider them on a par with any US amusement park. One thing I would add, we thought Tivoli would be a pretty quick visit. I was there in the early 80s and maybe there weren't rides, but we just didn't spend that much time. You look at it on the map and think, "It's only about a square block or so, surely I won't need to spend much time there." Au contraire my friend. We shut that place down! There is so much to see and do.
We utterly adored our time there. I hope you and yours do as well.