Discontinued Operations - Summary of the Significant Non-cash Items and Purchases of Property and Equipment (Details) - USD ($) $ in Thousands	9 Months Ended
Sep. 30, 2025	Sep. 30, 2024
Cash flows from discontinued operations operating activities [Abstract]
Share-based compensation	$ 55,733	$ 55,194
Community Living Business \| Discontinued Operations
Cash flows from discontinued operations operating activities [Abstract]
Depreciation and amortization	1,329	30,132
Share-based compensation	9,578	5,402
Impairment of long-lived assets	0	3,020
Cash flows from discontinued operations investing activities [Abstract]
Purchases of property and equipment	$ 8,755	$ 7,708

Cash flows from discontinued operations investing activities.

No definition available.

Name:	btsg_CashFlowsFromDiscontinuedOperationsInvestingActivitiesAbstract
Namespace Prefix:	btsg_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Cash flows from discontinued operations operating activities.

+ References

No definition available.

+ Details

Name:	btsg_CashFlowsFromDiscontinuedOperationsOperatingActivitiesAbstract
Namespace Prefix:	btsg_
Data Type:	xbrli:stringItemType
Balance Type:	na
Period Type:	duration

- Definition

Amount of capital expenditure attributable to discontinued operations.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 205
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 5B
-Subparagraph (c)(2)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483499/205-20-50-5B

+ Details

Name:	us-gaap_CapitalExpenditureDiscontinuedOperations
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Amount of deprecation and amortization expense attributable to property, plant and equipment and intangible assets of discontinued operations.

+ References

Reference 1: http://www.xbrl.org/2009/role/commonPracticeRef
-Topic 230
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 28
-Subparagraph (b)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482740/230-10-45-28

Reference 2: http://www.xbrl.org/2003/role/disclosureRef
-Topic 205
-SubTopic 20
-Name Accounting Standards Codification
-Section 50
-Paragraph 5B
-Subparagraph (c)(2)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147483499/205-20-50-5B

+ Details

Name:	us-gaap_DepreciationAndAmortizationDiscontinuedOperations
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

The aggregate amount of write-downs for impairments recognized during the period for long-lived assets held for abandonment, exchange or sale.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section 45
-Paragraph 28
-Subparagraph (b)
-SubTopic 10
-Topic 230
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482740/230-10-45-28

Reference 2: http://www.xbrl.org/2003/role/exampleRef
-Topic 820
-SubTopic 10
-Name Accounting Standards Codification
-Section 55
-Paragraph 100
-Subparagraph (e)
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482078/820-10-55-100

Reference 3: http://fasb.org/us-gaap/role/ref/legacyRef
-Topic 360
-SubTopic 10
-Name Accounting Standards Codification
-Section 45
-Paragraph 15
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482130/360-10-45-15

+ Details

Name:	us-gaap_ImpairmentOfLongLivedAssetsToBeDisposedOf
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Definition

Amount of noncash expense for share-based payment arrangement.

+ References

Reference 1: http://fasb.org/us-gaap/role/ref/legacyRef
-Name Accounting Standards Codification
-Section 45
-Paragraph 28
-Subparagraph (a)
-SubTopic 10
-Topic 230
-Publisher FASB
-URI https://asc.fasb.org/1943274/2147482740/230-10-45-28

+ Details

Name:	us-gaap_ShareBasedCompensation
Namespace Prefix:	us-gaap_
Data Type:	xbrli:monetaryItemType
Balance Type:	debit
Period Type:	duration

- Details

Name:	us-gaap_IncomeStatementBalanceSheetAndAdditionalDisclosuresByDisposalGroupsIncludingDiscontinuedOperationsAxis=btsg_CommunityLivingBusinessMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type:

- Details

Name:	us-gaap_DisposalGroupClassificationAxis=us-gaap_DiscontinuedOperationsHeldForSaleOrDisposedOfBySaleMember
Namespace Prefix:
Data Type:	na
Balance Type:
Period Type: