So, here's the breakdown I would offer (as DM) to a player. Others can double-check my numbers, but I think I got it all.
Level 1 Expert:
Size Small
AC 12
HP 9
Hit Dice 2d6 + 2
Speed 30
Proficiency +2
STR 12, DEX 13, CON 12, INT 4, WIS 10, CHA 6
Skills Athletics +3, Insight +2, Perception +2, Stealth +3, Survival +2
Saves DEX +3
Light Armor Proficiency
Features Displacement, Helpful
Attack +3
Damage 2 bludgeoning + 1 piercing
Level 2 Expert:
HP 14
Hit Dice 3d6 + 3
Features Cunning Action
Level 3 Expert:
HP 19
Hit Dice 4d6 + 4
Features Expertise (choose two skills for expertise, not included in skill bonuses below...)
Level 4 Expert:
Size Medium
AC 13 (same as Displacer Beast)
HP 30
Hit Dice 4d6, 1d8 + 10
Speed 35
STR 14, DEX 14, CON 14, INT 5, WIS 11, CHA 7 (add +1 to all ability scores, then apply ASI below)
Skills Athletics +4, Insight +2, Perception +2, Stealth +4, Survival +2
Saves DEX +4
Features ASI (STR +1, CON +1)
Attack +4
Damage 4 (1d4+2) bludgeoning + 2 (1d4) piercing
Level 5 Expert:
HP 37
Hit Dice 4d6, 2d8 + 12
Proficiency +3
Skills Athletics +5, Insight +3, Perception +3, Stealth +5, Survival +3
Saves DEX +5
Attack +5
Level 6 Expert:
HP 44
Hit Dice 4d6, 3d8 + 14
Features Coordinated Strike (or Multiattack, player's choice)
Level 7 Expert:
HP 51
Hit Dice 4d6, 4d8 + 16
Features Evasion (replaces Avoidance, or the DM could just let you get Avoidance)
Level 8 Expert:
Size Large (same as Displacer Beast)
HP 68
Hit Dice 4d6, 4d8, 1d10 + 27
Speed 40 (same as Displacer Beast)
STR 16, DEX 15, CON 16, INT 6, WIS 12, CHA 8 (add +1 to all ability scores, then apply ASI below)
Skills Athletics +6, Insight +4, Perception +4, Stealth +5, Survival +4
Features ASI (STR +1, CON +1)
Attack +6
Damage 6 (1d4+3) bludgeoning + 3 (1d6) piercing
Level 9 Expert:
HP 77
Hit Dice 4d6, 4d8, 2d10 + 30
Proficiency +4
Skills Athletics +5, Insight +3, Perception +3, Stealth +5, Survival +3
Saves DEX +6
Attack +7
Level 10 Expert:
HP 86 (+1 higher than Displacer Beast); can change to 85 (to match Displacer Beast)
Hit Dice 4d6, 4d8, 3d10 + 33; can change to 10d10+30 (to match Displacer Beast)
STR 18, DEX 15, CON 16, INT 6, WIS 12, CHA 8 (same as Displacer Beast)
Skills Athletics +6, Insight +3, Perception +3, Stealth +5, Survival +3 (Displacer Beast has no skills)
Features ASI (STR +2)
Attack +8 (+2 higher than Displacer Beast)
Damage 7 (1d4+4) bludgeoning + 3 (1d6) piercing (same as Displacer Beast)
This, for the most part, makes your sidekick a full Displacer Beast with some extras.