I will concede that I think the rules for lifting above your head is ridiculous; even I can lift more weight over my head than a really, really strong character in SR5, apparently.
That being said, I would allow the character to use just pure lifting off the ground which is multiplied by 15 instead of 5, for the sole reasons of activities like the caber toss and hammer throw. Imagine a troll grabbing a bike by it's handlebars, lifting it off the ground, swinging it around himself and letting go as he made a turn. That's more along the lines of what I was thinking off. Now, that probably wouldn't be possible for cars, but I'm thinking cars wouldn't be liftable anyway, with their weight of 800kgs and up I posited. A STR15 and BOD10 character can lift 225 + 15d6>5*15 kgs, and with a physical limit of around 15 the maximum such a character could ever lift would be 225 + 15*15 = 225+225 = 550kgs.
Wit that in mind, tossing a motorcycle like a hammer? Sure, I'd allow that. Picking up a small car? Probably not.
I would also as mentioned require two complex action (one for using a skill, even if it is attribute related, another for throwing an item), though I like the STR + Vehicle BOD damage factor. You could even use the vehicle BOD / 2 as a threshold to avoid getting pinned with a Gymnastics test if it successfully hit. Range is still something I'm unsure of; normally you would have penalties based on range, but STR + BOD (lifting, basically) - vehicle BOD * hits meters may make more sense as an absolute.
With the above, a STR15 + BOD10 + AGI 6 (just for laughs) with Throwing skill 6 character would roll 12 dice to hit, 25 - Vehicle BOD of 8 for a Harley = 17 dice for range totaling hits meters, doing potentially 23 physical damage if he succeeded. The opponent, if unable to dodge the incoming "weapon", would roll Gymnastics + AGI (4) to avoid being pinned underneat it, requiring an attribute Lift skill test to get out from under it.
I could work with that...