If you're dropping back on the shift from 6,200 all the way to 4,200 rpm (assuming non-lockup), then that indicates the converter's too tight. But if it's flashing to 4,000 rpm on the leave, then that's about right. I'd say something's off. With a glide, they typically say 1,200 rpm is about the max drop off you want. With a 3 speed (which is effectively what you're running), a little less because there's less spread between gears. If I read you correctly, you also say you're going through the traps at only 5,000 rpm? I'm going to assume a little over if it's locked up in 3rd, and maybe another couple hundred rpm if it's not.
I'd have to say I'd be surprised if that thing made peak hp above 5,500-5,600 rpm with that induction. 6,200 rpm might be a bit high to shift.
I think you're probably right to look into the converter. As has already been said, a good converter's not cheap. My first "good" converter was a PI Stallion 3,200 rpm 9.5" lockout for my AOD. I wouldn't do lockup for drag racing - the rpm drop on the 2-3rd shift is too severe. But that converter cost me $750 in the '90's yet. I currently run a custom BTE 10" converter set to full tight with a steel "C" stator. That one cost me over $1,000 and I had to send it back once to get it dialed in. But I'm putting that converter through hell - 1.80 low gear, 3.08 rear gear, 3,500 lbs and a 3,200 rpm transbrake leave at 13.5 psi. That's a lot of stress on the converter - they even told me not to use the brake with that low (numerically) of a rear gear ratio. Getting it to minimize slip at the stripe and only 6,200 rpm is tough. You'll probably want a smaller diameter converter. BTE doesn't make 4r70w converters that I know of. Edge (as mentioned) is a good source, so is PTE or Lenny at Ultimate Converter Concepts - maybe one of them will make you a converter for that trans.
But again, getting a converter that flashes decently with a good k factor (tq multiplication) behind your motor and still locks up decently down track is going to be tough with that low of an rpm ceiling. You'll really be better off letting that thing spin to at least 6,500 rpm, which means a new intake at least (that's where I'd start anyway). You'll probably end up with an 8" or 9.5" converter.
Oh, one more thing - don't get hung up on slippage while driving around. Yes, it does happen, but with a good converter it's not bad at all and hardly noticeable. Until you hammer it, anyway. Here's me tooling around with my converter on the street (stalls at 3,200 rpm on the transbrake/two step and flashes to 4,200 rpm and falls back to about 5,200 on the shift):
https://youtu.be/j8SNo7kpcxw
The only time I get into it a little is at 2:48 in.
Sorry for the long post. Hopefully all this helps.