Harris-Wright, 28, who had a season at Bristol in 2011-12, won a European Champions Cup medal at Leinster.
The club have also added Harlequins tight-head prop Tyler Gendall, 22, Jersey Reds back rower Nick Haining, 26, and Watsonians scrum-half Reiss Cullen, 20.
"We're pleased to bolster our options with these talented young players," interim head coach Mark Tainton said.

Witnesses told officers they heard a gun being fired near Anfield Road Primary School at about 09:00 GMT.
Pupils and staff there and at the nearby All Saints Catholic Primary School have been ordered to stay indoors while police investigate.
Ch Supt Mark Harrison said it "beggars belief" it would happen at a school.
Members of the public told police the shot was fired at a dark coloured car by a white man in a grey hooded top who was on foot.
A Merseyside Police spokeswoman said local hospitals had been checked and "no-one has sought medical attention for injuries caused by a firearm".
She said officers were making house-to-house inquiries and studying CCTV footage of the area.
Ch Supt Harrison said the attack "appears to be targeted" and was particularly worrying as "a child could have been seriously injured or worse".
He said police wanted to hear from those inside the car "so we know they are safe".
Anfield Road Primary School's headteacher Clair Drew-Williams said her pupils were "safe and unaware of the incident".
"The site was not evacuated, the school day is continuing as normal and a trip for some pupils took place as planned," she said.

Mr Blair said the party he used to lead should aspire to govern, rather than be a "fringe protest movement".
He has been a vocal critic of leader Jeremy Corbyn, warning before his leadership election victory that the party risked "annihilation" if he won.
Mr Corbyn's office declined to respond to Mr Blair's comments, made in a Spectator article.
"All wings of the Labour Party which support the notion of the Labour Party as a Party aspiring to govern, rather than as a fringe protest movement agree on the tragedy of the Labour Party's current position," he said in the article, which does not mention Mr Corbyn by name.
His comments come amid divisions within the party over its future direction. Most Labour MPs did not back Mr Corbyn for leader, but he overwhelmingly won the contest with over half of the membership's vote.
Tensions have surfaced since the Commons voted on air strikes in Syria. Some MPs have complained they were abused online after backing military intervention, and Mr Corbyn has warned members "abuse and intimidation" will not be tolerated.
There have also been claims so-called moderate Labour MPs could be unseated by pressure from the grass-roots Momentum group, formed from Mr Corbyn's leadership campaign, which the group denies.
In his article, Mr Blair also defended his party's record under his leadership, during which he said it had "effectively discarded" its ideological commitment to socialism.
Labour "fell short" in areas when hard choices had to be made, he said, with "significant elements" of the party unhappy with the "compromise" and "pragmatism" required for government, seeing it as a betrayal of Labour's principles.
Looking forward, Mr Blair said "more modernising and less ideological thinking" was needed to cope with challenges such as the housing shortage and social exclusion.
"Right now we're in danger of not asking the right questions, never mind failing to get the right answers," he said.
Many current Labour members feel the party "lost its way"  in government, Mr Blair concluded.
"I feel we found it," he said.
"But I accept in the process we failed to convince enough people that the true progressives are always the modernisers, not because they discard principle but because they have the courage to adhere to it when confronted with reality."
Bristol have re-signed London Irish hooker Jason Harris-Wright.
[NEXT_CONCEPT]
A shot was reportedly fired at a car outside a primary school in Liverpool as parents were taking their children inside, police have said.
[NEXT_CONCEPT]
The current state of the Labour Party is a "tragedy", former Prime Minister Tony Blair has said.