It might be cheaper as long as the OP doesn't have to need a new rear derailleur or hanger extender to accommodate a 36 big sprocket. But that still won't give as low a gear as a 22 small ring used with his existing cassette. For me a 9 speed 36 cassette would have bigger gaps between gear ratios than I'd be happy with, and I'd still want a lower bottom. I would prefer the overall lowering of the ratio range achieved with a different chainset. Often a triple 28/38/48 will be available in 44/32/22 at the same chainline which would not require a new bottom bracket.
But yes all valid solutions, depending on the needs/priorities of the OP. All perfectly doable and touring will be more enjoyable as a result!