You're essentially correct. There have been several attempts to explain away the inconsistency in Bird-of-Prey bridge design, of which the most common is on Memory Alpha:
Due to their age and variety of configurations, the main bridge design for Birds-of-Prey included numerous variations.
This is partly due to the fact that:
This [Star Trek IV: The Voyage Home's] bridge set was rebuilt several times, intermittently kept in storage, and underwent minor changes for each reuse.
But the immediate concern with the bridge design was money. Leonard Nimoy direct both Star Trek III: The Search for Spock and Star Trek IV: The Voyage Home, and was constrained by finances during the filming of the former:
According to director Leonard Nimoy, the set that would eventually be used for this bridge was originally in another movie as somewhere else and, when subsequently asked if the still-standing set would work as the Klingon bridge, he – under considerable financial constraints – felt pressured to answer positively. He also thought that – if there was enough money to redress the set as the Klingon bridge – the film's production crew could make the set look effective. After the bridge was consequently reconstructed as the Klingon bridge, Nimoy found the set to be very cramped, difficult, tight and limited in scope but he ultimately came to believe that the film's production team had managed to make it work.
The set was top-lit, which helped to hide the prosthetics of actors portraying the Klingons and make the applications believable. The film's production crew also tried to keep it dark, to give the bridge an ominous feeling.
The design used in Star Trek IV: The Voyage Home, was the one Nimoy wanted, and with the latter film's superior budget he, along with set designer Jack T. Collis, were able to achieve a design both preferred.
There has been an attempt to retcon the change in bridge configuration between the films as due to a re-fit on Vulcan:
It is possible that the differences in this ship's two bridge designs were the result of the reconstruction and repair of the ship during its three-month stay on Vulcan, in order to give the ship a more Starfleet feel. In reality, the change of bridge set was due to the fact that an entirely new set had to be constructed for Star Trek IV, following the demolishing of the Klingon bridge from Star Trek III. In addition, it was felt that the extra expense of building an all-new set was worthwhile, due to the fact that the Klingon ship would, in Star Trek IV, be featured to a much greater extent than it had been in the previous film. Leonard Nimoy found the new Klingon bridge set to be lavish by comparison with the previous one.
This retcon doesn't make sense, however, when one realises that this design is also used on other Birds-of-Prey during the course of TNG and DS9, including a slightly refitted version in the film Star Trek: Generations.