TLDR: per UN, 137k out of 306k civilian deaths came from "Government & allies"
The UN did a detailed report about this.
The report, mandated by the UN Human Rights Council, referred to 143,350 civilian deaths that have been individually documented by various sources with detailed information, including at least their full name, date and location of death. In addition, statistical estimation techniques of imputation and multiple systems estimation were used to connect the dots where there were missing elements of information. Using these techniques, a further 163,537 civilian deaths were estimated to have occurred, bringing the total estimated civilian death toll to 306,887.
They give you a link to the actual report, which is in a long list.
A/HRC/50/68 Report on civil deaths in the Syrian Arab Republic – Report of the Office of the United Nations High Commissioner for Human Rights English (Advance unedited version)
The thing is a docx. Looking for government, I eventually got to :
Table A6: Documented deaths by actors/groups alleged of causing the death and year
getting 137,529 as the government's "contribution".
They even provide some apportionment:
According to available data, 39.3 per cent of deaths (137,529) were allegedly caused by actions by the Government and its allies, 35.7 per cent (125,098) by non-State armed groups, which include anti-government groups (5.3 per cent, 18,519), Islamic factions (24.9 per cent, 87,039) and Islamic State (5.1 per cent, 17,868). 0.8 per cent of deaths (2,859) were allegedly caused by the coalition forces, and for 24.2 per cent (84,595) of the documented conflict-related deaths, the actors were recorded as unknown. It should be noted that to provide a more complete picture of the attribution of casualties to the various actors, more work would be required [...]
