Is It Safe to Leave the Temperature Wire Disconnected in Beats Headphone Battery Replacement?

Hi everyone.  I am replacing the battery in my bluetooth Beats headphones.  Seems pretty straight forward.   I was able to get the + and the - wire soldered just fine, but for the life of me I couldn't get the temperature wire to solder.  And now I have such little space inside the headphones, there is maybe 1/2" of the temperature wire left inside the casing, it is so tiny I just can't seem to get a hold of it, let along get enough real estate to solder it to the lead on the new battery.... 

Any ideas on fixing this or a work around?  Would that battery eventually explode or overheat if I just didn't connect the temperature lead?  The headphones are work as is, but I'm worried about that temp wire. 
In the pic, the arrow is pointing to the fray of the end of the little tiny bit of wire I need to connect to the new lead... That wire is so tiny and even has some insulation around it, it's almost impossible to even strip the wire. 


The battery is a 100 mAh 3.7v #481221.   The headphones are Power Beats Wireless 2.


Thank you in advance.

Difficult job. They don't make it easy, do they?  I think these were meant to be throwaway items, but I suspect you're like me, have a go at fixing anything...   I'd do your best to get the Temp wire on, it's there for a reason, and you don't want overheating batteries next to your ears.One technique you might try is to glue a couple of small bits of fine sandpaper to the ends of a pair of tweezers.  Applying gentle pressure you can sometimes strip corrosion, oxidation and even insulation from the wire leaving it clean and shiny so the solder will take.Be VERY gentle at first so you don't pull the wire off.Oh, and trim those red and black wires and INSULATE them.  Too late for heatshrink, but you can just use a fold of insulation tape.

Hi thanks for the reply.  Yes I am all about fixing stuff, I usually stick to woodworking though. These headphones I was just determined because my first pair bit the dust a few months ago and Apple warrantied them, which they swear up and down they dont send refurbs as warranty sets, but I don't believe that.  This pair has only been used about 5 months, but my warranty has expired so I'm SOL, and I figured it was something simple like the battery. 
Part of my problem too I think is my soldering iron, it's a small Weller butane one, and I worry about the heat vents on the side of it, they blow out enough hot air that I can't leave it in there too long for fear of melting plastic/nearby wires.   I do have a wood burner, but idk if that would work as a soldering iron though, and it's not variable temp.  Think I just need to bite the bullet and buy a small electronics soldering iron on Amazon.  Either that or i need some sort of heat shield, but there's just no room!
Sandpaper on the tweezers is a great idea, thank you.  I will try my best, I just worry I already buggered it up to much, but maybe getting that insulation cleaned off will be enough for it to stick.  And yes I was planning on cleaning up the wires before putting it back together. 
They certainly do not make it easy, it's ridiculous how tiny these wires are.   Thanks again.

Noah only one comment - get yourself a decent soldering iron.  I know the Weller Butane ones and they are great for working on vehicle radio installations and the like, but no good for that kind of work.  With the money you save on fixing stuff, look around for a decent iron!