The proposals, revealed on Tuesday by SNP Cowdenbeath MSP Annabelle Ewing will see a £45 charge introduced for broadcasting from Kirkcaldy and Dunfermline crematoriums.
However, the announcement caught administration councillors by surprise - including council co-leader Cllr David Ross (Lab), who wants the plans halted.
He said: "This is the first I have heard of it and if someone has made this decision then it needs to be reversed immediately.
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"At a time when there are tight restrictions on attendance at funerals with all the anguish this causes to those who have lost loved ones, it is ridiculous to think the council should start charging.”
The new charges were not detailed in any public documents, nor discussed at meetings of the council or its committees.
The issue is believed to stem from the original decision last May to begin livestreaming funeral services because of Covid restrictions that limited the number of people who could attend in person.
Council officers used delegated powers to introduce the scheme and agreed that the authority would absorb the charges for the rest of the financial year, which ends at the end of this month.
Cllr David Barratt, convener of the corporate services committee responsible for council-hosted funerals, has instructed officers to update the policy to continue the free streaming while funeral attendances remain curtailed.
He said: "Clearly, the hope had been that restrictions would have been lifted by the new financial year, but when the circumstances that led to the decision to provide the service persist, it is right to extend the measure and to offer live streaming free of charge for as long as restrictions are in place.
"These were operational decisions that only came to my attention this week having been raised by Annabelle Ewing.
"As convener of the relevant committee, I have asked for this charge to be removed immediately.
"The council has received money to mitigate the impact of Covid restrictions, and I expect the costs associated with this service to be met by that and not from mourning families."