Huh

Why would they block it?
This all depends on how updated the BIOS is. As you can see from the support page
here, it looks like there's been 5 BIOS updates for your board.
For the latest chips including the Pentium anniversary you'll need the latest bios version 0904, download here -
link
This was released on 26/5/14 so unless you've very recently got the board it's unlikely to have this BIOS revision installed.
You could try buying the chip, it may POST and if so the chances are you'll be able to update it to the latest BIOS, but obviously this is a risk and by the sounds of it a risk you can ill afford to try.
You could always buy a cheaper CPU like the Pentium G3220 that's been supported since the first BIOS and then update from there, although that kinda defeats the object of keeping costs down.
I'll be honest with you the anniversary Pentium is really suited to a Z97 board that will really get the most out of it when overclocked as that's what it's been designed for. It seems a bit of a waste to slap it in a H81 based board.
Like others have mentioned you'd probably be better off looking for an older i3 chip like the 4130 or 4350, at least you'll know these will definitely work as they've been supported from the first released BIOS.