ok jake, without seeing your tree i cant say anything concrete, but if you are tied to the adjacent tree and the first one falls youre going to swing into the tree youre tied to, if youre using some kind of break away youll likely be shot into the tree youre tied to and in not aware of any "break away" carabeaners
without seeing it i cant say for sure but if its standing its probably safe to climb, live pine trees catch a lot of wind, so if the wind hasnt done it in, youre not likely too, as you climb youll be limbing and removing weight likely more then half your own weight
as you limb it dont leave stubs, cut flush to the tree, stubs have the potential to create chaos
one or two at the top for a hand hold or lanyard stop arent a bad idea
the higher you go before you top it the better, the more weight you suddenly free up from the top of that tree the more violently the trunk is going to buck, and once youre down to the pole, if youre up high enough its going to be pretty wobbly, the limbs and top act as stabilizers
stay steady in the tree, keep your weight centered and use yourself to counter the wobble, this takes some practice
when youre climbing spend some time inspecting the weak area of concern
if you decide its unsafe to climb, dont climb it, tell the customer you think it needs a boom, possibly call in another more experienced climber for a second opinion
sounds like they know youre a beginner and are working with you, no doubt your personal safety is a great concern for them
youre plan to be shot or swung into an adjacent tree isnt something i would even consider
if you truely think the tree will break from your added weight, dont climb it
but again, if its still standing, youre probably safe
do you plan to just tip the top off or does it need to be roped? if youre roping it, without the proper equipment and someone who knows how to use it youre going to shock load it
this is going to send you for a ride, keep youre weight on your spikes, _know_ what youre going to do with youre saw and be ready to hang on, make sure youre secured in a way that if your spikes are kicked out you dont ride your lanyard down the trunk
this is also where the largest risk of the tree breaking is
if youre just tipping it off, depending on how much flex is between you and the ground, how much weight youre taking from the top and how cleanly your notch breaks free, youre still gunna be bucked up in that tree
just be sure youre ready and youre sure how and where the top will fall
thats about as good of advise i can give you