I'm slightly reluctant to suggest this because I haven't tried it in perspex but ... would you get a good enough finish by parallel drilling and then making a home-brewed taper reamer? No idea what relief you would need on the reamer, but it should be fairly easy to make one and experiment.
If I had to do this I would drill first with a 4.7mm or so drill and then ream with a home made "D" bit reamer. I have made several form reamers this way but none with such a high ratio of length to diameter. I just checked and the one I have made that is the closest to what you need is about 7mm at the big end, about 4mm at the small end and only about 35mm long. I make mine from drill rod, which is what you probably call silver steel. Don't use any cutting oil when machining acrylic because it will eventually cause cracks. I learned this the hard way. Water works OK. There may be something better. I'm not sure how you will support your reamer to mill away half the diameter. I use a low melting temperature alloy (170F) for this kind of work. The stuff I use, after cooling for an hour, exactly fills the cavity it is poured into. So for this job I would mill a small stepped cavity in an aluminum or steel bar that the reamer would fit into. The higher step holds the shank and the lower step allows a little room for the alloy. Then the reamer is clamped to the higher step and the low temp melting alloy is poured into the area needed to support the part to be cut. I pour enough to go slightly above center, but not much, maybe .5mm. Mill the tool about .025mm below center. Remove and either heat treat and then stone the milled surface or with Perspex and only one hole you may get away with no heat treat of the tool. I would try no heat treat first. Just stone the flat so the edges get really sharp. When reaming you will of course need to remove only a little at a time so the tool will be entering the hole many times. Oh, I forgot to mention, the reamer needs to be turned to the proper taper before milling. You probably already know all about "D" bit reamers, but if not, or if someone reading this does not, then maybe I have passed a little knowledge along. I hope so. Thanks for reading to the end. Cheers, Eric, Who lives on an island named after Royal Navy Officer Joseph Whidbey.
As others have said, a reamer. I believe musical instrument makers make their own tapered reamers to bore out woodwind instruments, and these are considerably longer than your requirement (and subtly curved). Possibly if you find a discussion group or website used by such folk you may find some suitable hints.
I would try making a tapered bar of the size you need then heat the perspex and push in the bar to stretch the perspex. If you can polish the bar it should help you get a nice internal finish. I haven't heat formed perspex recently so can't remember a suitable temperature but when I last heat formed some PVC drain pipe for a mould I found 165C was a good temperature. I expect you can find some information online. Lubricant might help but obviously something that won't attack the perspex.
When stuck recently for a tiny boring bar I remembered a tip I picked up years ago to use one flute of and end mill. I suitably mounted the end mill in a holder with it's axis slightly tilted to the direction of travel and the cutting edge on centre height and it worked superbly.
I did something a bit like that, but just used the tapered reamer without any lubrication, in the lathe, so the reamer got hot - I had been assiduously keeping the tool cold before, but letting it get hot worked much better.
The surface isn't great, but I expect I can polish it up a bit, first with some fine wet-and-dry then toothpaste.
Your taper is 1:60 on diameter. A metric taper pin reamer will give 1:50 and an imperial one 1:48 on diameter. reaming 60mm deep in Perspex would be no problem at all. On that size, steel isn't a problem. The only worry is whether you would get any micro-cracking in surface.