Tall order,that: IEM's under $150 that sound closest to E5's. Sorry,but I think you won't get close at all with that budget. E4's with UM56 tips will get you quite close to E5's with anything (to my ears) in the stock Shure fit-kit applied to stock E5's. And going that route will also provide slightly brighter,more detailed treble in comparison with the stock E5 tips on E5's,while bringing up the slightly recessed bass on the 4's in comparison to stock Shure tips.
I've spent several weeks now swapping out E4 and E5's (both fitted with UM56 inserts) through several dozen songs I know very well,through two sources (Zen Micro,Zen Sleek) amped (AE-1) and unamped. Won't bore you with some soundcard comments through a couple of different computers using WAV.files as first link.
Perhaps saving a bit until you can order either UM2's or E4's with the custom UM56 inserts will satisfy your ears and your wallet in your quest to approach the ear candy your E5's provided you? I think Shure did a modestly amazing job in engineering the E4,and I also think it highly unlikely that you'll find anything in your current budget ballpark that will match or exceed it; add the premium bump for UM56's plants you squarely in UM2 terf on price-point.
So...What to do? Plan ahead and/or budget more and be patient? Yup.
Good luck,whichever path (spend your budget right now,or save for later) you take.