You should be handling it in practice like you would in real life. Meaning, if you do not have time to do all the engine failure actions AND still get set up for the approach, you prioritize. Handle the emergency, then call ATC and request vectors back around for the approach if that's what you need. That shows good CRM versus trying to rush everything.
Even with an engine failure on takeoff, it's better to do the actions slow and accurate than fast and incorrectly.
I wanted to expand a bit on this now that I'm at a real keyboard.
I used to do a lot of multiengine training. Without fail, the first time I'd fail an engine (which would be up at cruise altitude), the student would react instantly and start trying to throw all kinds of levers and switches right away. Invariably they would miss steps, get the wrong levers and switches in the wrong place, and just generally forget everything we talked about and ran through on the ground.
There's the old saying about the first action to take in a emergency is to wind your watch. Not because the winding of your watch is important, but because it slows you down, which allows you act properly and take deliberate action instead of reacting improperly to try to get everything done quickly.
Since almost nobody winds a watch anymore, I modify that to be basically, "take a deep breath, mutter the expletive of your choice, then get to work running the actions". I often will demonstrate this by failing the engine, taking the breath, saying something nice and slow and relaxed like "oh darn, there goes that engine again" and then running the action items - noting that the airplane didn't immediately enter a death spiral during my brief soliloquy.
This is true even on an engine failure on takeoff, where it seems counterintuitive. But it's far better to do the RIGHT actions slowly than the WRONG actions quickly. And really, we're talking just 2 or 3 seconds here - that's all it takes to get your brain out of the panic/react mode and into the deliberate, considered action mode.
Of course, as they get more experience and practice, they get quicker anyway.
I do exactly this as a pilot during my twice-annual recurrent training and checking (simulator and airplane). "Oh dang, looks like the left engine is on fire again. Okay, left condition lever - fuel cut-off, left prop lever - feather, left firewall fuel valve - close, fire extinguisher - actuate. Engine fire checklist please."