It's to my understanding that the PM held the position for life. Could they attempt to further their political career? Could they participate or lead in the army?
I'm not sure when it ceased to be a lifelong position, but Julius Caesar became Pontifex Maximus in 63BC. You may be aware that he later rose to higher political office.
Publius Licinius Crassus Dives was made PM in 213 or 212 and served until his death in 183. He served as Consul in 205.
Marcus Aemilius Lepidus was made PM in 180 and served his second Consulship in 175.
Publius Cornelius Scipio Nasica Serapio became PM in 141 and was elected Consul in 138.
Publius Licinius Crassus Dives Mucianus became PM in 132 and then Consul in 131.
Gnaeus Domitius Ahenobarbus became PM in 103 and Consul in 96.
Quintus Caecilius Metellus Pius became PM in 81 and Consul in 80. But he may likely be excluded from your consideration. As he was a Sulla diehard and got both positions due to that. So that is definitely in the period where constitutionality of such things wasn't much of a concern.
And there may have been others.
But no, unlike the Rex Sacrorum and the flamines, the Pontifex Maximus was not bared from holding other political offices. In general it was a position that was given to a senior statesman. So it was generally a cap to a career. But it wasn't that unusual for a PM to hold other political offices.