It looks as though your cousin’s “injuries” are only to his pride—both “in the doghouse” and “throwing someone under the bus” are vivid idioms, not literal events. In everyday English they just mean (1) he’s in trouble or out of favor, and (2) you shifted the blame to him to save yourself. So unless there really was a wayward city-bus careening through a backyard kennel, his bones and fur are perfectly safe—only his feelings might be a bit flattened. Below is a quick crash-course on both expressions, plus a few tips for patching things up.

“In the doghouse”

Meaning

To be in the doghouse is to be in disgrace or disfavor with someone whose approval you normally enjoy—most often a partner, family member or boss. It conjures the image of being banished outside with the dog until you earn forgiveness.

Where it came from

The first clearly figurative use shows up in U.S. slang dictionaries of the 1920s (“in dog house, in disfavor”) and gained traction through the 1930s press. Popular lore also credits Peter Pan (1911) for spreading the image of Mr Darling sleeping in the kennel after a parenting blunder, giving the phrase extra cultural punch.

“Throw (or toss) someone under the bus”

Meaning

To throw someone under the bus is to sacrifice, blame or betray them for self-preservation or advantage, as though pushing them into traffic while you stay safe on the curb.

Origin snapshot

The earliest solid print example so far is from 1982 in The Times (London), with wider U.S. political use exploding by the 2008 election cycle. Its exact birthplace is still debated, but all roads lead to late-20th-century journalism and politics.

What this means for your cousin
Tips to get him out of the kennel-zone
Own your part – Admit you “drove” the bus. A candid apology often shortens kennel time.
Repair, don’t excuse – Help fix the original problem rather than focusing on who got blamed.
Set future guardrails – Agree on how you’ll both handle mistakes next time so nobody ends up canine-camping again.
Inject a little humor – Shared laughter about the idioms can defuse tension, reminding everyone the “injuries” were metaphorical.

Key sources consulted

Phrases.org.uk: first printed definition of “in the doghouse”

Wikipedia entry on “Throw under the bus” for meaning & early citations

English StackExchange & TheIdioms.com for origin discussions

Paula Reed Nancarrow blog on Peter Pan link to the doghouse image

Bottom line: your cousin’s ego may need a pat on the head, but his vitals are almost certainly intact. A well-timed apology (and maybe a chew-toy peace offering) should get him back on the porch in no time.