Assuming that you don't want to have ilfochromes done (you still can! but they are very pricey), then the next best thing would be to have drum scans done of the slides, then use lightjet or chromira or similar output. Still further down in cost would be to have non-drum scans done, and some scans from the Nikon machines, for example, can be very good (albeit not good enough for 16x20 from 35mm slides, in my opinion; for those enlargements, I'd stick to drum).
If you have a lot of slides then you might consider purchasing your own Nikon scanner. I do not think anything less than a Nikon will do a good enough job on 35mm slides, though I did once have a Minolta scanner (now discontinued) that did quite well. A flatbed will not cut it with 35mm slides enlarged as much as you say, in my opinion. You'd have to overscan massively and it'd not be worth the filesize.