Blackmailing is 'do this or I'll release bad information about you'. I'm not really sure how that would apply to such a situation. Do you mean coerce? I agree that it wouldn't be looked upon well, but only because Mathilde is so good and useful; Belegar pays money to have Mathilde on call, so of course it wouldn't be reasonable to expect that he'd keep paying us if we weren't on call, but it wouldn't be coercive to request a sabbatical, only relatively blind to the realities of the Karak's situation.
Of course, at the end of the day, it's all in how you phrase it. 'Pay me to take a vacation' is bad, and worse when you really need somebody at the workplace or else things might literally burn down without them. 'As a master of my craft, I'd like to retire from this position, but still be located nearby to help out the Karak, of course, in order to delve into the greater mysteries that could make me a truly worthy grandmaster' is quite reasonable in comparison, as long as it comes at a time when there aren't eight fire-hazards in the Karak itself. Belegar knows how useful furthering our mastery is in the abstract sense; some real grandmasters just came down to Eight Peaks and built him an apocalypse weapon that utilizes the flames of dwarven hell.