It is meant to remind Britons of the “example set by our ultimate public servant”, but a £4.4m government scheme to send out free portraits of Charles III for display in public buildings is not quite going to plan.

After first limiting the availability of the oak-framed pictures of the king in his admiral’s uniform to courts, schools and police and fire services, the deputy prime minister, Oliver Dowden, said this week that any jobcentre, university, Church of England church or hospital could also have one.

In doing so, the government managed to rile workers in the stretched public services, with university lecturers calling it “culture wars nonsense”, and mosques, who accused it of excluding other religions.

It is unclear if the decision to widen the scheme is a sign that take-up has been lower than anticipated, although the Cabinet Office did say the extension would not cost any more than the original budget. That could suggest the original contracted supply has not been exhausted.

It declined to say how many portraits had been requested in total, choosing to issue only selected uptake figures for court buildings (100%), principal local authorities (93%)and Home Office buildings (80%). It said it was “pleased” with the first phases of the scheme.

With 31% of UK adults saying they would prefer an elected head of state, according to YouGov polling in January, the initiative has divided opinion and is proving a test of how far deference has changed since pictures of the late Elizabeth II were hung widely.

HM Coastguard said its staff would be “proud” to display Charles’s portrait in operations centres and search and rescue helicopter bases. The National Association of Civic Officers said the image now hung in many mayors’ parlours and council chambers. But unions representing nurses, welfare workers and lecturers greeted the offer with disdain, and members of the public have given it a mixed response.

Jo Grady, the general secretary of the University and College Union, representing academics, said the offer was a “descent into parody”. “British higher education needs sustainable public funding and an end to government attacks on academic freedom, not subsidised worship of the king,” she said.

The GMB union, representing jobcentre security guards embroiled in a pay dispute, called for the money to be used to lift wages instead. “It would shame the king were he to know that his portrait was now looking down on hard-working men and women only earning the minimum wage,” said Eamon O’Hearn, a national officer.

Jon Richards, the Unison assistant general secretary, said: “It’s clear the royal family is a source of comfort and national identity for many people, but not everyone will view this as a good use of public money. Universities, hospitals and other essential services would happily find better ways for this cash to be spent.”

And the Mosques and Imams National Advisory Board (Minab)said Dowden’s decision to single out C of E buildings as the only religious locations eligible for a freebie “excludes other religions and communities”.

The Cabinet Office stressed that the C of E had a unique relationship with the state. But Charles, when he was Prince of Wales, vocally supported the rights of all faiths, telling the BBC in 2015 that “while at the same time being defender of the faith, you can also be protector of faiths”.

Qari Asim, the Minab chair, said: “King Charles III is held in a very high regard by people of all faiths because he has deep interests in faith and the work that he has done to build bridges in communities. Since he has been in his role as king he has been engaged with faith leaders. Various faiths would have liked to have shown that respect is reciprocated by having his picture hung up.”

The image of the king in a gilded room at Windsor Castle was taken in November by the photographer Hugo Burnand. It is less dramatic than Jonathan Yeo’s first official painted portrait of Charles as king, which was unveiled this week. Yeo depicts him facing the viewer from a sea of scarlet and magenta, which more than one viewer has said suggests the king is in some sort of inferno.

The scheme was originally expected to cost £8.2m. In October, the government contracted a private company to print, frame and deliver the A3-sized portraits for £4.4m. In January and February, it offered the pictures free of charge to councils, courts, schools, police forces, fire and rescue services and cadet forces.

Six weeks ago, Alex Burghart, a Cabinet Office minister, said the government had no plans to extend the offer to places of worship, but said they could buy a picture.

This week, Dowden changed position and offered the portraits to hospitals, coastguard operations buildings, jobcentres, universities and C of E churches. The Cabinet Office said there was no warehouse of unused portraits because they were being produced in small batches on demand. Buckingham Palace declined to comment.

One Guardian reader who got in touch about the portraits said they welcomed the “unifying sense of pride it creates”. Another in Buckinghamshire said she and her friends giggled while warming up for their kickboxing sessions in the parish council room in front of the king.

A teacher in Wiltshire said: “We have no money for things we actually need – like a roof that doesn’t leak – but there is money for a painting of the king.”

Another teacher in Yorkshire said: “I ordered it when I heard that public money was being wasted on such an outrageous vanity project. We keep it in an office away from the students and put a different face on it every week. This week it’s William Shatner.”