Sure, it's possible - I already said I had flown with a guy on our server years ago who used only mouse and keyboard, I simply do not know how he achieved it. I will note that he considered himself Combat Mission Ready when we went out on a sortie, but was unacceptably slow with threat reaction and target engagement - I assume due to the lack of a HOTAS setup. Controls and input must be done in a swift and timely manner, and mistakes are punished harshly. There is a reason our Air Force pilots do not have a keyboard and mouse in the cockpit, because of the number of commands which need to be sitting at the finger tips ready to be activated at a moments notice without the pilot taking his hands off the throttle and stick (HOTAS). Like I said, many of these essential controls are listed in Jeffu's blog.
This is basically the real deal, real life things not modeled or supported are few and far between, and that short list includes nuclear consent SOP's, classified IFF SOP's, Air Flow/Oxygen requirements of the pilot, and of course the tremendous stresses on the physical body (G forces) are non-existent (except for simulated screen effects when sustaining too many G's). If they put a remote control unit in the jet, everything you do could be translated to the real thing.
Just because these guys above have no solutions for operating BMS with only a mouse and keyboard does not mean they are not CMR experts - well, everyone in this thread but Vanilla and m823us - we all have thousands of hours in this sim, and between the three of us, have helped hundreds who are just starting out in this hobby for nearly a decade (here at VG, anyway). There is no shortage of experts over at the BMS Forums, either, and your chances of getting advice on an established method for mouse/keyboard flying would be higher over there, being the main hub for this simulator. No shortage of "blind leading the blind", too, so watch for date of joining, number of posts, and number of thanks. BlueWolf might be the first to reply with a proper method, I assume - or maybe Arty. *You will get MANY MORE suggestions to purchase a simple joystick, just be polite and remember they do not know you've already been over it many times already. DO not feed the trolls.
Again, best wishes and good luck!