The name is genuine, but it does not belong to the very early Iranian material that was crystallised by Ferdowsī in the Šāh-nāmeh.
Shād-u-kām / Shādkam / Shādkamān (شاد و کام ، شادکام ، شادکمان ; literally ‘Joy-and-Fulfilment’) is a place that occurs in the later “romance” or dāstān tradition – stories that were composed in Persian and, still more frequently, in Ottoman Turkish and Indo-Persian/Urdu between the 14th and the 19th centuries.
In those romances it is usually described as one of the provinces of Jinnistān, the fairy-land ruled by the Perīs (fairies) and Jinn.

1. The Hamza Romance (Qiṣṣa-yi Ḥamza / Dāstān-i Amīr Ḥamza)
   • Persian prose versions are already attested in the 15th century (e.g. BNF ms. Persan 526, dated 862/1458).
   • In the opening chapters the sorcerer-emperor Afrāsiyāb sits on the Ruby Throne in Jawhar-ābād, the capital of «Shādkamān, vilāyat-i pariyān» – “Shadkaman, the domain of the Peris”.
   • An easily accessible English rendering is Musharraf-Ali Farooqi (tr.), The Adventures of Amir Hamza, Modern Library, New York 2008, pp. 3-4:

    “In the kingdom of Shādkamān in Jinnistān, Afrāsiyāb the Accursed … held court in the jewelled city of Jawharābād.”
   

2. Tilism-i Hōsh-rubā (Urdu sequel of the Hamza cycle, composed c. 1880, printed Lucknow 1883-94)
   • Re-uses exactly the same geography: the magical tilism is created “in the fairy province of Shādkamān”.

3. Qaḥramān-nāmeh / Kahramannâme (Ottoman Turkish, 15th–16th cent.; hero = Qaḥramān, Persian Narīmān, father of Sām)
   • Ed. Mertol Tulum, İstanbul 1977; modern critical ed. Necati Demir, Ankara 2012.
   • BnF ms. Suppl. turc 825 (16th c.) fol. 35b: “vilâyet-i Şâd ü Kâm, ki der memleket-i Perîyânest” – “the province Shad-u-Kâm, which is in the land of the Peris”.
   • Here the two kings of Shād-u-kām ask Qaḥramān for help against the Dīvs – exactly the episode Keightley summarised.

4. Miscellaneous Indo-Persian fairy romances
   • ʿAjāʾib-nāmeh, Baḥr-ul-Gharāʾib, Dāstān-i Malak-Muḥammad, etc. – all contain short excursions to “Shādkam” or “Shādkaman” and repeat the stock description of its luxuriant gardens and jewelled palaces.

What is NOT a source

• Ferdowsī, Šāh-nāmeh – the toponym never occurs in the standard text.
• Earlier Middle-Iranian (Pahlavi) literature – no trace.
• Classical Arabic geographers – no trace.

Why the confusion?

European mythographers normally worked with the 18th- and early-19th-century French translations of the Hamza romance (Garcin de Tassy, Langlès) and with the first printed Turkish Kahramannâme (Constantinople 1833). Keightley simply pulled the passage from one of those versions; Eco in turn cites Keightley.

Where to read it today

1. Kahramannâme (Ottoman Turkish)
   Demir, Necati (ed.). Kahramannâme. T.C. Kültür Bakanlığı Yayınları 2012.
   – see canto II, vv. 421-540 for the description of Shād-u-kām and its capital Jawhar-ābād.

2. The Adventures of Amir Hamza
   tr. Musharraf-Ali Farooqi, Modern Library 2008.
   – pp. 3-10 give the whole scene in Shadkaman; the word is spelt exactly so.

3. Urdu Tilism-e Hosh-rubā
   tr. Musharraf-Ali Farooqi, Hosh-ruba: The World-seizer, Random House India 2009, ch. 1.

In short

Shād-u-kām is not part of the “canonical” Iranian mythosphere but belongs to the later popular epic of the Persianate world. Its oldest extant appearance is in 15th-century manuscripts of the Amīr Ḥamza cycle; it is described again, with more detail, in the Ottoman Turkish Kahramannâme and many 18th- to 19th-century Indo-Persian fairy romances. Those texts are the primary sources that Keightley (and after him Eco) ultimately relied on.