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 a medium difficulty encounter for a 3 or 4 person level 3 party. Adding a bugbear chief to your party adds a much stronger fighter who will outshine other martial characters. Damage output is significantly higher: two attacks per round and 2d8+3 damage per attack that hits. A regular bugbear looks like a better fit than the chief, but an adjustment here or there may be helpful.
Compare a Fighter, level 3, to a Bugbear (CR 1).
(From page 120 of Basic Rules(2018)):
CR 1. Armor Class 16 (hide armor, shield) (Compares favorably to fighter at lvl 3); if you give him scale mail and shield, the AC is 17.
Hit Points 27 (5d8 + 5) (Close to an average level 3 Fighter of 28: 10 + 6 + 6 + 6 (Con 14 (+2)x3)
Speed 30 ft. (Same as a standard level 3 human Fighter)
STR 15 (+2) DEX 14 (+2) CON 13 (+1) INT 8 (−1) WIS 11 (+0) CHA 9 (−1)
(Below the standard array, legal via point-buy)
Skills Stealth +6, Survival +2 (Comparable but weaker; +6 has some situational advantages, but most PCs have 4, 5, or 6 Proficiencies; this is strictly a downgrade)
Senses darkvision 60 ft., passive Perception 10
(A bonus versus human, same as a dwarf or elf, comparable, perception weaker).
Languages Common, Goblin (No significant difference, weaker than a Half Elf)
Here's where it gets tricky: each monster has different special abilities. This one's got two combat 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.
In tier 1 play, this represents a major difference in damage dealt per round versus a level 3 Fighter who uses sword and shield, and is comparable (slightly more) with a Fighter who uses two handed weapons. The sword and shield 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. Bugbear won't get surprise every round either. Power comparison is close enough.
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 shield Fighter. Compare to a 2H sword (2d6 + 3) versus 2d8 +2; close. That reach has some, occasional, tactical benefits in more opportunity attacks being within the Bugbear's reach. For example, an enemy who is adjacent to the bugbear's ally is still within the Bugbear's OA range when they move away from the ally - so that enemy potentially gets two OAs: one from the bugbear, one from the other party member.
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 (2d6+2 vs 1d6 +3) 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. That Bugbear, who looks pretty good in Tier 1 play, falls behind quickly once tier 2/level 5/two attacks for a Fighter comes online and gets an ASI or an additional feat.
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 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. However, the PC bugbear will have more skill proficiencies, and will get all of the class abilties like fighting style, second wind, action surge, etc, that this "straight" Bugbear won't have.
A couple of years ago, our party ended up leading an orc revolt against their hill giant masters, and one of the orcs became a follower of our party's Champion. We eventually fitted him out (per Volos's) as a PC, and he survived a few levels as an Orc PC who took Fighter as Class and Battle Master as archetype. By the time he got to level 5 and was using Chain and Shield, he was way more powerful than any orc we'd run into and could give an Orc Chieftan a run for his money.