Is it bad if a devotee of Lord Shiva would like to worship Lord Vishnu at the same time?
Yes. Any devotee of Shiva can worship Vishnu and any devotee of Vishnu can worship Shiva. All our scriptures including the Vedas and the Puranas(both 'sattvik' and 'tamasik') boldly declare the equality of Lord Shiva and Lord Vishnu.
Equality of Shiva and Vishnu
Vyasadeva says to Shiva in Shanti-Parva of the Mahabharat(Reference : Omkarnath Rachanavali,Mahamilan Math, Vol.11, page 39)
You played with the cowherd boys in your incarnation as Krishna and You are Govinda, the Omniscient One.
The Varaha-Purana says in chapter 25
purusho vishnur iti ukto shivo vaa naamatah smritah/avyaktam tu umaa devi shrir vaa padmanivekshanaa// meaning : The Primal Being is known as Vishnu or Shiva in Manifest form and is the lotus-eyed Umaa or Shri (Lakshmi) in the Unmanifest form.
The same purana in the Sauvaagya-vrata chapter reads
yaa srih sa girijaa proktaa yo harih sa trilochanah/evam sarveshu saastreshu puraaneshu cha pathyate//etasmaad anyathaa yastu brute saastram prithak taya/tam naastikam vijaaniyaat sarva-dharma-vahishkritah// meaning : She Who is Shri (Lakshmi) is Girijaa(Paarvati) and He Who is Hari (Vishnu) is Trilochana(Shiva).All the scriptures including the Puranas say this.The person who sees them different is an atheist and rejected by all 'dharma'-s.
The Vishnu-Puraana says in the 5th part of chapter 3:
matto abhinnaatmaanam drashtum arhasi samkara/yo aham sa tvam jagat cha idam sdeva-asura-manusham/avidya-mohitaatmaanam purushaa vinnadarshinah//(48). Here Sri Vishnu is saying to Sri Shiva : I and You are one and the same in the essence--the All pervading one residing in all gods, demons and men.Only the persons deluded by ignorance sees us differently.
The equality of Brahmaa and Vishnu/ Shiva is also mentioned in many scriptures.The Padma-Purana says (svargakhanda, chapter1, sloka 51,63)
brahmarupadharo devah svayam eva harih parah/..raamaadirupam sa tu grihya paati babhuva rudro jagad etad hartum// Lord Vishnu takes the form of Lord Brahmaa for creation in Rajo-guna, preserves the dharma taking forms like Raama and destroys the world in the form of Rudra.
We hear the same about the equality of them in the messages from Srimad-Bhagavatam also.
Worshipping of Vishnu by the Shiva Devotees
It is clear from the above that any real devotee must worship both Shiva and Vishnu. One of course may love more the Ishta form, but he can not think the other form is inferior as per the scriptures.
For the devotees of Shiva, tye worship of Vishnu is special also for two reasons:
Shiva Himself chants the name of 'Rama' as mentioned in the Kashi-khanda of the Skanda-Purana.
In Purana-Samgraha, Lord Shiva says
anarham mama naivedyam patram pushpam falam jalam/saalagrama-shila-lagnam sarvam yaati pavitrataam// meaning that the food, leaves, flowers, fruits and water offered to Lord Shiva must be accepted only after getting them touched to a Shalagrama (Vishnu) Shila.
One interesting incident
Madhav Swamiji, a very close disciple and sevaka of Sri Sitaramdas Omkarnath, Who belonged to the Ramanuja-Ramananda-lineage, took sannyasa from Sri Lakshmana Ramanuja Jear, the pontiff if the Srirangamm Falahari Jear Math. Once both of them were staying with Sri Omkarnath at Bhubaneswar.
Omkarnathji told to Madhavji: Let us go to the Lingaraj Shiva temple for darshan.
Madhav Swamiji (Kinkara Madhava Ramanuja Jear) replied: No Master, I am forbidden by my sannyasa-guru to visit any Shiva temple.
Omkarnathji gave him a smike and said : My child! Have I to remind you again that Hari and Hara are one?
--I am sorry, Master. I am not allowed by my rules of sannyasa.
Omkarnathji held one hand of him and went to Sri Lakshmama Ramanuja Jear straight.
Lakshnpmanaji respecrpted Omkarnathji like his own Guru.
Omkarnathji just told him: Come with us. Wevare going to have a darshan of Lord Shiva at the Lingaraj temple.
At once Lakshmanji bowed down to Him and held His hand. The three joyfully went to the Shiva Temple and worshipped Lord Shiva.
Sri Lingaraja is known as the Hari-Hara linga by the way.
Omkarnathji writes in His 'Sri Sri Shiva-Mahimamrita' (Omkarnath Rachanavali,Vol.11, page 25)
'There are some Vaishnavas who say that Shiva is inferior to Vishnu--They are not ready even to offer a pronam to Shiva. What is the matter?
Those who say so are far from realisation of the Truth as yet'.
No, most definitely Not.
A devotee of Lord Shiva may definitely worship Lord Vishnu at the same time, and vice-versa too.
The Hari-Hara non-difference has been affirmed by Shiva himself to Kumara Kartikeya the Skanda-purāṇa.
Chapter 33, Kaumārikā-khaṇḍa, Book 1 - Māheśvara-khaṇḍa, Skanda-purāṇa
महादेवो ह्यथालिंग्य स्कन्दं वचनब्रवीत् । यद्भवान्मम भक्तेषु प्रकरोति कृपां पराम् ॥ ४३ ॥ तेनापि परमा प्रीतिर्मम जाता तवोपरि । किं तु यद्भगवानाह वासुदेवो जगद्गुरुः ॥ ४४ ॥ तत्त्था नान्यथा किंचिदत्र प्रोक्तं हि विष्णुना । यो ह्यहं स हरिर्ज्ञेयो यो हरिः सोऽहमित्युता ॥ ४५ ॥ नावयोरंतरं किंचिद्दीपयोरिव सुव्रत । एनं द्वेष्टि स मां द्वेष्टियोन्वेत्येनं स माऽनुगः ॥ ४६ ॥ इति स्कन्द विजानाति स मद्भक्तोन्यथा न हि ॥ ४७ ॥
43-47. Mahādeva then embraced Skanda and spoke these words: “Since you show great kindness towards ray devotees I am extremely delighted with you. But what Lord Vāsudeva, the sire of the universe, has said is so and not otherwise. What has been said by Yiṣṇu cannot be altered. It should be known that I am Hari and Harī is me. O (Lord) of holy rites, there is no difference between us as in the case of two lamps. He who hates him, hates me. He who follows him is my follower. He who knows this, O Skanda, is my devotee. Not otherwise.”
English Translation by G.V. Tagare