First of all I'm just going to say T-Cut and hand polishing won't break through your clear coat, unless it has previously been poorly resprayed. I use much heavier cutting compounds when polishing more sensitive areas of cars both by machine and hand, so I really wouldn't let that enter into your head.
If you have a cutting compound like t-cut sitting in the garage use it (After giving it a good shake), it really depends how badly the bird has etched the paint as to whether it will remove it. Definitely apply wax or sealant after it, that's what helps to protect your car from birds!
If you can't get it out, do mention it to a local detailer that has a fixed premises, it doesn't have to get expensive if you don't want it to! £20-30 would fix just that which is a lot cheaper than constantly looking at it and it annoying you

Also the price of polishes and cutting compounds now are generally expensive for good ones, if I recommended you one you'd be well past £20 for a bottle so it'll probably be cheaper to get a local detailer to do it!