You could try a MK2 with a balanced/adjustable 2nd stage. That might set you back about $100 and it would be very freeze-resistant, and ultra reliable. The comment about "beware of cheap scuba gear", while well-intentioned, is not really accurate.
Most contemporary scuba gear is priced on features, hype, and shelf-appeal. Pricing has nothing to do with safety.
The MK2 is not sealed, and while a fully sealed 1st stage would be ideal for ice diving, the MK2 does have a great cold water reputation because of it's relatively low flow flow-by, not flow-through, piston design. Stepping up from the MK2 would be either a MK17, or any number of sealed diaphragm 1st stages. I believe that the old balanced/adjustable is probably about the most freeze-resistant 2nd stage available, but you'll have to find one used.
If you're talking strictly about new regs, the MK17/G20V would be a good choice if it's in your budget. Certainly there would be absolutely no need to spend anything more than what that costs.