Labours welfare spokes-woman Jackie Baillie said she hopes the government will step in to provide more cash to mitigate the effects and then her Bill would not be needed.
However, she said her determination to proceed should not be under-estimated. The Bill would mean no-one could be evicted for arrears arising purely from the benefit cut from the bedroom tax.
The Bill was backed by anti-poverty campaigners but opposed by housing associations who fear a loss of income and a detrimental effect on the ability to provide maintenance.
Ms Baillie said: "While pressure from Labour has already forced the Scottish Government to offer some money to mitigate the effects of the bedroom tax, it is clear that this is nowhere near enough.
"The cost of mitigating the bedroom tax is estim-ated by Shelter to be in the region of £50 million each year. The SNP must find the resources within its £30 billion budget to fully mitigate the cost of the bedroom tax and protect council and housing association budgets."
Charities including Children in Scotland backed the Bill.
In a submission to the consultation the charity said: "Whilst acknowledging the pressure on limited social housing stock, forcing families from previously allocated housing for this reason can have negative consequences in terms of removing children from their school, friends and other support services and networks, with attendant impacts on their quality of life and development."
The Scottish Federation of Housing Associations opposes the Bill, stating: "It is never in the interests of a housing association to evict a tenant without having pursued all other alter-natives. Scotland's housing associations do not pursue eviction lightly, but they are a necessary last resort."
Ms Baillie added: "I will now be taking forward my final proposal for a Bill and I hope that, with support from MSPs on all sides of the Chamber, the Parlia-ment will agree to legislate to ensure that no-one is forced to leave their home due to the bedroom tax.
"There is an easier way, the SNP could ensure the resources are on the table and there would be no need for the Bill to proceed."
The Scottish Government announced last year it would be providing cash to allow councils to top up the discretionary housing payments (DHP) it receives from Westminster.
Finance Secretary John Swinney said in his budget that an extra £20m would be available this year 2013/14.
Then in October, Deputy First Minister Nicola Sturgeon announced it would be made available the following year too.
Labour says £50m is needed, but Mr Swinney has said the cash allocated is the maximum allowed under Westminster rules.
Labour however wants other mechanisms to be explored to provide cash to tenants to make up the shortfall in benefit payments.
It suggested a prevention of homelessness fund where cash from the Scottish budget could be provided to councils.
In Glasgow the cash available to tenants for DHP this year is £6.4m with £3.5m from the Scottish Government added to the £2.9m from the Department of Work and Pensions.