Are we sure this motor uses hydralic lifters? Anyway I think lifter noise is very short in duration, like 20 seconds or so until oil pressure gets oil to the lifters.
Since this sound lasts a few minutes it is either a bad manifold gasket, or a cracked exhaust manifold. I have a similar noise with my '01 Toyota Sequoia and I’ve elected to live with it. If the noise gets louder, or lasts longer, or never goes away, then I’d take some action. It has been stable and unchanged for 5 years so it is now just “normal” for the car at this stage of its life.