I would probably treat him as being mounted on the roc. He is effectively trying to do the same thing that being mounted on it would do. You can't wait until your mount moves you and then make a full attack.
If he insists on using the rules for grapple instead, then the roc can only move at half speed while grappling him. Also if the roc ascends at a 45 degree angle it moves at half speed. So if the roc flies up while grappling him it would move at quarter speed.
The enlarged roc probably would have no trouble lifting him as it likely had a strength over 30 and its carrying capacity is multiplied X 4 for size. Add that to the X 3 bonus from ant haul and the thing can pretty much lift entire buildings.
Isn't a dire tiger much larger than a normal tiger? So wikipedia couldn't help much, go with the bestiary stats.
If you allowed him to take flyby attack with it (requires his roc to have 3 INT), then he can make the grapple while en route to his destination. The roc would move at full speed so long as it isn't flying up at a 45 degree angle until it grapples the druid and then would drop to half speed for the rest of its move so long as it doesn't fly up at a 45 degree angle (then it would be quarter speed).
Sidenote: Flyby attack isn't legal in PFS as it is a bestiary feat, not sure if this is a PFS game but just saying.
Without regard to the Roc pikcing up the tiger and how long it would take or if it can carry the tiger lets assume he gets to the point in the air where the tiger is 10 ft above his enemy.
I can't recall if a falling speed is ever actually stated anywhere in Pathfinder, but I believe in some specific spells and such is specifies falling at 500 ft/per round. This is something that just happens and requires no aciton on the tiger/druid's part. So he just falls the 10ft. The first 10ft of falling only causes 1d6 non-lethal damage. And if you don't take lethal damage you don't even have to make a check to not fall prone.
Falling
Creatures that fall take 1d6 points of damage per 10 feet fallen, to a maximum of 20d6. Creatures that take lethal damage from a fall land in a prone position.If a character deliberately jumps instead of merely slipping or falling, the damage is the same but the first 1d6 is nonlethal damage. A DC 15 Acrobatics check allows the character to avoid any damage from the first 10 feet fallen and converts any damage from the second 10 feet to nonlethal damage. Thus, a character who slips from a ledge 30 feet up takes 3d6 damage. If the same character deliberately jumps, he takes 1d6 points of nonlethal damage and 2d6 points of lethal damage. And if the character leaps down with a successful Acrobatics check, he takes only 1d6 points of nonlethal damage and 1d6 points of lethal damage from the plunge.
Falls onto yielding surfaces (soft ground, mud) also convert the first 1d6 of damage to nonlethal damage. This reduction is cumulative with reduced damage due to deliberate jumps and the Acrobatics skill.
A character cannot cast a spell while falling, unless the fall is greater than 500 feet or the spell is an immediate action, such as feather fall. Casting a spell while falling requires a concentration check with a DC equal to 20 + the spell's level. Casting teleport or a similar spell while falling does not end your momentum, it just changes your location, meaning that you still take falling damage, even if you arrive atop a solid surface.
So, the tiger/druid would fall 10ft (possibly take 1d6 non-lethal damage or make a check to suffer none) and then have his full round worth of actions that he can take. This is not a charge or a pounce, but it is a full round attack. For cats (tigers) this may be important because he might not get the additional hind claw attacks or what have you.
Edit: Since it's a full round it looks like he would be able to get his 2 claws, bite, and rake if the 2 claw attacks are successful.
From the title, I thought this thread would be about Implant Bomb. Such disappointment. Imagine: Pounce, multiple attacks and then BOOOM!
As for the original question: I'd allow it. But throw in an acrobatics check or this would fail miserably. Then I would point out that if the "tiger" had fly cast on it, this would not be an issue.
among Smilodons 6000 lbs would be unheard of. Wikipedia suggests that the largest of the smilodon species capped out around 800 lbs. Larger than a normal tiger for sure, but nowhere near the 6,000 lbs bulk that the Dire Tiger entry suggests.
The Smilodon text is giving you an appropriation of what the animal looks like (with the saber fangs and all) I don't think its meant to describe size.