I would, let him down.
He is only 19 months old. At this age, "expectations" has to be age-appropriate...otherwise it will be frustrating for both parent and child.
A child this age, cannot just "sit" even if toys are placed in front of them. Ever have to go to the Opera with your Parents as a child? Well, imagine that BORED out of your mind feeling... and wanting to leave... but your parents saying you have to sit & stay, not squirm and act like a well behaved "adult." Well, depending on the age of the child, it AIN'T going to happen.
There is a time, and a place, and an age... at which certain things are not attainable yet. Per their development. Sitting "nicely" at a dinner table, at home or at restaurants, especially at this age through about 4 years old... is hard. Some kids may be able to do it... but not for 1-2 hours straight. Nor for 1/2 hour straight...which for a child is the same thing as 1-2 "hours."
Him climbing on your lap... well, kids do this. Both of mine did that too. Sure, it can be annoying sometimes... but well. The thing we did was take turns eating, or I just STILL ate my meal with them on my lap... explaining "sit on this side, Mommy needs to eat..."
Or, instead of a High-chair.. why not just give him his own "big boy" chair"? I, never really used high-chairs... my kids just sat at the table with us, in a regular chair. Or, they had their own "kid sized" table, to sit at.
In time, as he gets older, it will get easier. Sure, kids need to learn "rules" and what not..... but still, if developmentally a kid cannot attain something, then forcing it won't make it any easier.
As Deanna Leigh said: kids this age DO NOT have "fully developed" attention spans... BUT, nor do they even have fully developed "impulse-control" or emotions, or fully developed full spectrum logic based reasoning yet.
My son just made 3 years old... he is a great kid and eats at the table nicely on a regular dining chair. BUT... being a kid, he still likes to come by me, or on me, to "eat." He just likes to be near me. If I were a marsupial... then he'd probably sit in my pouch everywhere I went. LOL. I asked him "why" he likes to sit on Mommy... and he says its because he loves me, and that I am so "cozy." Aw shucks...
But yes, you also need to introduce "rules" about nursing... my kids both self-weaned. But for my daughter she was about 2.5 years old when she self-weaned... anyway, long story short is I "taught" her RULES about nursing... ie: no pulling/tugging at me NOR in public, asking politely, no demanding, not in public etc. And she understood and had more "manners" about it.
Well, all the best,
Susan