If you apply a thin line of paste across the pads, if you are lucky, the inter pin shorts will fix themselves as the solder flows onto the pads.
But seriously, just a cheapie hot air gun is all you need.
I have an old kada 858D that someone gave me 8 years ago and it had had a decent amount of usage when I got it, it's still going strong. You can get these or the "modern day" equivalent for not much at all.
It just needs to blow air and have adjustable fan and temp.
Heck, if you have a hot air gun, you can use that!
For something like the chip you are doing, I'd put some solder on the pads and a little bit on the centre pad(you don't want too much), hot air to get the solder melted, flux on the ic, hold it near the hot air stream to preheat, then when the solder on the pads is melted, drop it on and poke it about with the tweezers to be sure the solder has flowed properly.
Go up to the bean counters office and take all their modern equipment and then tell them to hurry up and get their jobs done
Seriously, not having a hot air station in you line of work is just plan stupid.
You have put more time into thinking of alternate ways to get that chip on the board than what a hot air station costs and you could have had it on the board before you had the time to make your first post about it.
Sometimes I'm sure management just likes to be difficult because they are wankers!