First of all, it looks like either your saddle is too small fore you or your stirrups are much too short. Your knee extends over the knee flap area. Try dropping your stirrups two notches. This will also help you move your foot down under you some, instead of being so far out in front of you in a chair seat.
The hrose moves with what looks like a "defensive" posture, that may be from an illfitting saddle. That may be one reason why it gets worse when you trot. Have you cantered him? Is it even worse then?
Yes the horse needs to have some contact on the bit and some give in front but if he is moving this way to try and protect his back then it is unfair to ask him to reach down.
If the saddle does fit well (I really question that, but we might be able to help you on that) then your next step is to encourage this pony to start reaching downward and outward to help stretch out his back and to get him confident in reaching for and meeting the bit. This means that you walk him along and using a kind of tickling with your hands on one rein, the inside rein, you "invite" him to reach forward. The instand he does, you give a big release on the rein. Do this over and over and you can even say "down!" and ask him to reach a bit more each time before you give the release, and eventually he will reach forwrad and downward, while you maintain contact to his mouth (Which is your challenge). If you can get this and maintain contact on the mouth and keep him walking forward with some push from behind, this is an excellent foundation excersize.
I wish I could be there to show you, because it looks like a really nice place!