I main a Mesmer and it can be played to fit every criteria you're looking for - not to mention it looks very "battle-mage-y" with a greatsword.
You could go with a power/precision/shatter build for the bursty damage you're looking for. Greatsword, albeit traditionally a melee weapon, is ranged for the Mesmer class and synergizes well with the Sword if you want to get in close for a burst and then dodge-roll back to get into a far range.
Someone said Mesmers are pretty squishy. I'd agree with that for the earlier levels, but once you get the hang of clones/phantasms, and most importantly the Deceptive Evasion trait in the Dueling line, you'll be able to mitigate damage through your illusions. Mesmers get plenty of Vigor everytime a clone crits an enemy - so you should be able to dodge constantly. Combine that with ranged Greatsword and you shouldn't need to worry about being squishy. Or just go with some Knight's armor and Beserker's jewels and you have a good balance between toughness and power/precision.
Guardian is a good runner up in my opinion - but it will definitely feel less like a caster.
On the earlier levels, I used the clones to distract the mobs so they don't attack you - so with the utility skills of Mirror Images (summons two clones) and Signet of Illusions, your clone production and their longevity will increase in the earlier levels.
Shatter them when you see their health dropping around 50-25% to get some added damage.
Later on, the clones help stack bleeds as well with the trait Sharper Images, so they become handier - and with Deceptive Evasion, you'll almost always have 3 clones up - which is great for PvE.