There is an old saying: "Buy once, cry once". There are some nice barrels that can be purchased for $100 from Green Mountain, Whistle Pig, ER Shaw, etc. I guarantee that if accuracy is your main goal you will buy another barrel later. Meaning you will buy twice and cry twice. I suggest you dig deep into your wallet and just buy a Volquartzen or Kidd the first time. You will be much happier from the start. Besides, a great bull barrel from Kidd is only $100 more than what you want to pay.
My suggestion is this. You are looking to spend $300-$350 for a barrel and trigger ($100 barrel and $200 trigger), right? Because most of the good triggers are $200+. Instead I would suggest you spend $200 on a Kidd bull barrel and $105 for their trigger job trigger and install it yourself. Now you have spent the same amount of money and have the most accurate gun possible with a great trigger. This is what I do on my 10/22 builds. I bring my 10/22 rifles to the range quite often. If you would like to try them out I will be happy to let you put them trough their paces anytime. And if you would like some help installing the Kidd Trigger Job Trigger I will be happy to help you with that as well.
A couple of other notes. The 10/22 stock is not capable of fitting properly with a bull barrel. So you will need to find a suitable replacement stock that is capable of fitting with a .920" diameter barrel. Also, take the previous suggestion and read all you can about 10/22 mods on Rimfire Central. It is the best resource I have found.