You shouldn't add an extra beam to the tongue if the rest of the trailer frame isn't already designed for that type of tongue.
I don't know what this particular BBQ smoker trailer looks like, but most of your common small trailers are basically a piece of truss with wheels on it. The long beams on the sides are the "chords" of the truss and the shorter beams that run across and support the floor/cargo are its "web."
An A-frame tongue has the tongue attached to the chords, but if you add a third beam to the tongue and weld it onto that front cross-member, unless that cross-member is designed to be loaded that way, it will probably make the trailer frame more likely to fail, instead of making it stronger.
There are two reasons why. One is kinda obvious, you are putting some of the vertical tongue load on the front web piece, and it might not be meant for that much load.
The other reason is not so obvious, you are adding pushing/pulling loads to the side of the beam caused by accelerating and braking the trailer. If that front web piece is C-channel it is almost certainly not meant for having a tongue piece welded onto it because you would be loading it from the side of the beam (front of the trailer), and this can bend it very easily.
I'm not an engineer but if I was going to upgrade a trailer tongue I would just replace the whole tongue, not weld a new beam to a place on the trailer frame that isn't meant for it. $0.02.