I can't speak anything regarding whether you should do it or not. It is the relation between devotee (you) and Lord.
However after seeing your description of your question, I felt to write this answer.
I got this random thought whilst reciting Shiva Chalisa...
With this statement, I remembered Grihapati incarnation of Lord Shiva which is discussed in Shiva Purana (Shatrudra Samhita chapter 13) as follows:
O gentle lady of beautiful hips ever desirous of my welfare, what is there that cannot be granted to you? O lucky one, you may ask for it. I shall grant it to you without delay.
O good woman, by the favour of Lord Śiva, nothing is inaccessible to me either here or hereafter. Lord Śiva is the cause of all good.
On hearing these words of her husband she who thought her husband to be a deity was delighted. Joining her palms in reverence she humbly said.
O Lord, if I deserve a boon, if a boon is to be granted to me, please give me a son equal to Lord Śiva. I am not choosing any other boon.
On hearing her words that Brahmin of pure rites entered into mystic trance for a short while and thought in his heart thus.
“What is this? This lady has wished for an extremely inaccessible thing. It is beyond the reach of wishes. Let it be. Only He does everything.
It is by Shiva himself stationing himself in her mouth in the form of speech that this has been said. This alone is possible. Who else could have endeavoured to say so?”
After thinking like this, the sage Viśvānara of liberal heart who strictly adhered to the principle of monogamy spoke to her.
Obviously we can't compare ourselves with the characters and devotees mentioned in the Purana. But if the thought spontaneously arose in you while reading Shiva Chalisa, then it could be the desire of Shiva himself.
Shiva's method of rewarding devotees are very weird sometimes. Sometimes one may transgress scriptural rules also and still they won't get punishments. Like in Skandha Purana (Purvardha chapter 13) there is a story which goes like this:
On seeing the attendants of Sambhu, the servants of Yama became frightened.
After bowing down they said: "Oh Ganas, this Brahmana is vicious in his habits. He violated the noble practices of the family. He was averse to the words of
advice of his parents. He swerved from the path of truth and purity. He remained
without holy ablution during the Sandhyas. Let his activities be kept apart. He took away the Nirmalya (remnants of
adoration) of Siva which you have seen directly. He is not worthy of being touched
by people like you. The very touch of these persons causes loss of merit: those who enjoy Siva's
Nirmalyas, those who transgress Siva's Nirmalya and those who give away Siva's
Nirmalya. One can stir and drink poison; observing fast is more conducive to welfare,
but Siva's possession should not be resorted to even when the very vital breaths
are near the throat. You are the ultimate authority regarding piety. We are not so. O Ganas, if
there is a vestige of piety in this man let us hear about it."
On hearing these words, the attendants of Siva said: "O servants (of
Yama), the pious practices of (a devotee of) Siva are very subtle. How can they be
viewed by you of gross vision? They can be viewed (and realised) by only men of
subtle vision. Listen to what has been done by this sinless one.
The shadow of the lamp falling on the top of the Linga was prevented by him,
for he added a wick to the lamp at night from the corner piece of his own cloth. O servants (of Yama), another great act of piety has been added to his credit
by him, as he listened to the names of Siva repeated casually by someone.
The worship that was being performed duly by a devotee was witnessed by this
man with steady mind, who had observed fast on the fourteenth lunar day. This excellent Brahmana whose sins have been dispelled shall now become the
king of Kalinga. O messengers (of Yama), you all may go in the manner you have
There is a story of devotee called Rudra Pashupati. While he was hearing Puranas he heard the story of Lord Shiva drinking poison which come from churning of ocean. He suddenly becomes very emotional hearing that Lord Shiva drank poison and begins to cry. Even Lord Shiva came to give him darshan but he was just interested in removing that poison he saw on the neck of Lord Shiva. He began to argue with Lord Shiva to spit out that poison and remove that poison from the neck. Such was his love. He just started to saw Lord Shiva as a child.
So, the relation of Bhakta and Bhagvan is established on the basis of emotions. Lord can accept various things if done out of devotion or compelled by certain emotions towards love.
But saying this I'm not endorsing that you should put Shiva Linga in the refrigerator. Actually if the Prana Pratishta is done in Shiva Linga then it can remove the energy of Linga itself by doing such things. But still saying all of these the devotion is relation between Bhakta and Bhagavan.