Shooting a 'standard' trap gun (high shooting) at non-rising target does take some pactice, but it can be done. Many shooters do prefer a high shooting gun for everything, and I'm one of them. I use the same trap gun, wobble or not.
To me, 'lively' means a light barrel set. Beretta, Perazzi, etc... the guns move when told, and are good for games with lots of gun movement - wobble, crazy quail, ZZ.
The 390/391 sport is in this category also... many people shoot those things scarry well. Excellent dynamics for 'find and chase' games.