Energy (calories) from climbing is dead simple. m*g*h. Kilograms times gravity times meters climbed equals joules. By a mathematical and physiological quirk, kilojoules is about the same as nutritional calories used.

This is additional due to climbing, plus whatever you'd have burned if it was flat. Hard as it may be to believe, simple as that.

The body going into starvation mode is known to be more or less a myth nowadays. The dip in metabolism people are concerned about is transitory. You're just hungrier and eat more, due in part to an increase in some hormones, and the base metabolic rate is slowed somewhat immediately after severe dieting.