one of the razer blackwidow ultimate models was released with cherry blues and blue backlighting and is within your budget.
however.... i would never buy a razer keyboard personally since they were plagued with problems and the new versions (green backlighting) are tied to razer synapse which is a complete pos software.
i am not aware of any full sized mechanical keyboard for $100-125 with cherry mx blues and blue backlighting that i would feel comfortable recommending.
you can likely find some foreign made brand or no-name brand but again... how good are they?
you may be able to find a tenkeyless keyboard with the options you want however they will still likely run you $100+ and that is not for a full sized keyboard.
i spent $120+ for a non backlit pretty basic cherry mx board from wasd. the deal from maxkeyboards really is not bad at all. there are cheaper boards on the market for sure however i've read of a few issues with a few of them and many dont offer backlighting. if they do it is typically pretty uneven. maxkeyboard has one led under each key which is as good as you are going to get. it may cost more but the devil is in the details.
mechanical keyboards tend to be rather budget unfriendly for anything worth recommending. is there a particular reason you want mechanical over rubber dome? you can get a rubber dome keyboard easily in your budget with blue backlighting.
if you must have mechancial then honestly i'd suggest raising your budget a bit, settle for no backlighting or settle for a cheaper brand of keyboard which may or may not perform up to your expectations. of course you are more than welcome to continue looking and post what you find here for us to look at and let you know what we think. i just havent seen anything out there that fits your budget, illumination and keyswitch needs which i would personally buy let alone recommend.