OP not sure if bought a barrel now or not...
Don't buy a ER Shaw yet!! If you do, don't get from Midway you could get a threaded muzzle one from Shooters Discount for $13 more... But yeah don't get an ER Shaw, and here's why:
There's a review on that ER Shaw, on midwayusa, that I 100% agree with. They have a very short/non-existent chamber leade. Incredibly tight Bentz chamber. Bullets will engrave upon simply chambering a round. This would shave off some of the bullet even while riding the slide. Imagine what firing it was doing lol. Mine was over 75% reliable with good 38/40gr HV (solids were best) ammo and the BX-25 magazines. Ran like a champ for 200 rounds or so then the FTF or FTE. I of course use a VQ exact edge extractor so that wasn't to blame. I blame the overly-tight Bentz chamber and rifling. These are 4140 steel construction. Mine had largely non-existent and forgettable crown finishing work. Otherwise barrel is good and will perform (as long as kept clean with good solid point ammo) just as well as the next $100 barrel
Anyways since you said under $150 you might also consider one of these KSA Micro Groove Barrels. These are 4150 steel. Finish is nice. Have one 16.5" Blued fluted for $130. The fluted are a good shake lighter than that solid ER Shaw. KSA also makes a new 3 flute version of this barrel not listed on the above website, but rather only their ebay. It's even lighter and more expensive of course
From there might as well make the jump to a Kidd Ultra Light Weight Fluted for $185 lol Personally I like some weight to my rifle so the KSA was perfect and the 4150 material is nice but wholly unnecessary in this caliber. But at least you can throw that number around in internet talk to appear tacticool lol. This barrel is also micro grooved: having 12 lands and grooves. Some might say that's less upset of the delicate 22 bullet during firing, leading to more accuracy. Whatever. I could not say. What really caught my eye is the fact that it uses a 'Ruger Target chamber' unlike 95% of other aftermarket barrels using a 'Bentz' or variation thereof. The Ruger target of course is a much more forgiving and reliable chamber, and my gauges prove it. Less accurate some may argue, but I don't shoot $0.38 ELEY EPS bullets all day in this thing so i wouldn't know anyways. The KSA has run 1500+ with no cleaning only 4-5 failure to extracts. IMO this barrel will meet/exceed the accuracy of the ER Shaw and will do so more reliably. That at least is my experience. YMMV