If you want to try alternatives, look for demi-sec sparkling wines. "Demi-Sec" means "half-dry" but compared to brut it's quite sweet. "Brut" effectively means "extra extra dry" and typically has almost no residual sugar. "Extra dry" has a little sweetness, "dry" has even more sweetness, and "demi-sec" is an overtly sweet wine, like Moet Ice Imperial. Long, long ago, Champagne was all very sweet (even sweeter than demi-sec), but over the years a dryer style has become more popular.
Total Wine has an inexpensive house sparkling brand called Depreville and their demi-sec also says on the label "serve over ice." It's only about $15/bottle so it may not be as complex as Moet, but a lot more affordable as a summer picnic sparkler. There are other options out there and any demi-sec is probably worth at least a try over ice, and you might even find a "dry" or "extra dry" sparkler that's sweet and robust enough to put over ice. Depreville also makes a demi-sec rose sparkling wine that may be worth a try. Schramsberg makes a highly-regarded California demi-sec sparkling wine that might be very good with a little ice.