You can get cheap RC lipo batteries which give 2S~6S (series) li-po that can handle current nicely (they do not, however, provide over-current protection). My suggestion would be get one that has more than 12~13V drained, use a buck converter with UVLO set at drained voltage + safety margin and use that to power your system. Anything that requires boost instead of step down will require higher current from batteries, and anything in between will require buck-boost which is a bit less readily available in breakouts.
So, e.g.:
4S 5500mAH Li-Po (you can get ones with less capacity cheaper)
4S = 4 batteries in series, which gives ~3.7V*4=14.8V safely discharged and 4*4.2V=16.8V charged.
You can then use a readily available LM2956 step down converter (although you can find more efficient or smaller options) to step your 14.8~16.8V to 12V.
This reduces a bit the current requirement (current, not power!) and should be an easier load on the battery.
There is one problem with RC batteries though. They tend to not provide over-current, over-draining, thermal, etc protections, so I'd recommend integrating an UVLO circuit to your setup and possibly a fuse.
Page 27 of LM 2956's datasheet shows a simple schematic and description of how to implement this (note that Z1 in your case would probably have to be ~14V):
So, how long would it last?
Say your setup consumes 12V * 0.6A = 7.2W
Quick google around says that at 3.7V a LiPo cell has 30% left, so we can use 70% of battery power.
LM2956 efficiency is about 85~90% for 15V to 12V:
So:
5500mAh*14.8V*0.7*0.8/7.2W = 6.3h
Which seems quite enough to me!
Then depending on your uses you could buy two smaller batteries instead of a huge one, etc etc