The short answer is that a regular bugbear (CR 1) is a closer add on to the party -- even so, he's a fighter who might outshine other martial characters at level 3, and is at least comparable.
The other answer is No, there isn't a ratio that is easy to use
Note: At low levels, the relationship gets tied to difficulty level of an encounter, and the ratio may appear to be three or four to one, but it's inexact. I'll illustrate why with your Bugbear example.
A monster’s challenge rating tells you how great a threat the monster is. For example, a party of four 3rd-level characters should find a monster with a challenge rating of 3 to be a worthy challenge, but not a deadly one. (Basic Rules, DM, p. 5)
Your CR 3 Bugbear chief is an encounter for a 3 or 4 person level 3 party, so he's closer to equivalent to 2 or 3 3d level characters. Adding a bugbear chief to your party adds a much stronger fighter who will outshine other martial characters. That's not such a great idea. A regular bugbear looks like a better fit than the chief.
Compare a Fighter, level 3, to a Bugbear (regular, CR 1).
(From page 120 of Basic Rules(2018)):
CR 1. Armor Class 16 (hide armor, shield) (Compares to fighter at lvl 3)
Hit Points 27 (5d8 + 5) (More than a level 3 Fighter)
Speed 30 ft. (Same as level 3 Fighter)
STR 15 (+2) DEX 14 (+2) CON 13 (+1) INT 8 (−1) WIS 11 (+0) CHA9 (−1)
(Not quite standard array, legal via point-buy)
Skills Stealth +6, Survival +2 (Comparable)
Senses darkvision 60 ft., passive Perception 10
(A bonus versus human, same as a dwarf or elf, comparable).
Languages Common, Goblin (No significant difference)
Here's where it gets tricky: each monster has different special abilities. This one's got two substantial advantages.
Brute. A melee weapon deals one extra die of its damage when the
bugbear hits with it (included in the attack).
Surprise Attack. If the bugbear surprises a creature and hits it with
an attack during the first round of combat, the target takes an extra
7 (2d6) damage from the attack1.
Major difference in damage dealt per round versus a level 3 Fighter who goes sword and board, and comparable (slightly more) than a fighter who uses two handed weapons. The sword and board fighter may have higher AC, and a comparable armor if going with the two-handed weapon / Chain mail set up. (Minor differences come out in the wash assuming a fighter with +5 (+3+2 prof) to hit and the bug bear with +4 to hit (+2 +2 prof))
The Rogue-like bonus when attacking with surprise. While a Fighter will have Action Surge, it doesn't trigger every round, though the Bugbear won't get surprise every round either. (A wash).
Morningstar. Melee Weapon Attack: +4 to hit, reach 5 ft., one target.
Hit: 11 (2d8 + 2) piercing damage.
That's more than the 1H (notional 1d8+3) weapon for a sword and board fighter. Compare to a 2H sword (2d6 + 3) versus 2d8 +2 ... comparable.
Javelin. Melee or Ranged Weapon Attack: +4 to hit, reach 5 ft. or range 30/120 ft., one target. Hit: 9 (2d6 + 2) piercing damage in melee or 5 (1d6 + 2) piercing damage at range.
Same as your fighter at range, more effective in melee with the same weapon.
You state that ...
Balance and challenge rating is still very important,
Final answer: you need to do a detailed analysis (something like the above breakdown) of any monster you'd want to add to your party, and compare it to a PC, to see how close the power level is. CR is a soft guide at best.
1 If the party can set up a surprise attack/ambush, the Bugbear can do 2d8+2 + 2d6 in that round where a fighter would do 1d8+3 or 2d6+3 -- but most rounds of combat won't have that advantage.
WoTC has provided an "easy button" now on pages 118 through 120 of it's most recent official material, Volo's Guide to Monsters. The section entitled "Monstrous Adventurers" includes the same sort of starting racial mods for a bugbear as for dwarf, elf, halfling, etc, on page 119. It resembles the Goliath in terms of increased carrying capacity (one size category up), a boost in Strength and (in this case Dex), sneak attack damage like a rogue, and some other mods.