I have to check on my CO leadership skill during creation. Then I will place chits for LAM and pilot.
You're currently capped out on current personnel. You'd have to let someone go to make room for the new pilot. Not clear if you can fire people during a contract. You could still get the LAM and then use the standard recruitment rules for a LAM pilot out of the v5 rules.
"LAM Pilots: Make an Acquisition Roll to acquire a LAM pilot; the TN is 10. You may attempt 1 Roll per month, and may not roll if in transit. This attempt takes 5 of your monthly acquisition "slots". Your roll gets a bonus of half your CHA score (round down), no other modifiers except for Bad Reputation and Contact apply. If your MoS is 0, you find a Green LAM Pilot (5/6 to both Mech and ASF piloting stats). If your MoS is 1-2, you find a Regular LAM pilot (4/5 Mech scores, 5/4 ASF scores; 1 random SPA on a 9+). If your MoS is 3+, you find a Veteran LAM pilot (3/4 Mech Scores, 4/3 ASF scores; 2 random SPAs each on an 8+). You MUST hire the pilot in question if you find one, and may not fire them for at least 6 months. Add +$50,000 CB to their hiring bonus. All qualified LAM pilots are automatically classified as both Medium and Light Mech pilots."
You could wait until you bring in a new CO to try and recruit a pilot.
Alternatively, if you get a pilot killed you'd have a spot. It it's your CO, you could even conceivably take the LAM pilot as your new CO if they're cool.