# [Charcoal], <s>115</s> 112 bytes

    ≔⟦⟦⟦⟧⟦⟧⟧⟧ηＦＥθ÷×÷⊖Ｘ⁴ι²Ｘ²κＸ²ι¿ι«≔⟦⟧ζＦη«≔§κ¹εＦΦＬε¬＆§ελι⊞ζ⟦⁺§κ⁰⟦λ⟧Ｅε⎇⁼νλ⁺μιμ⟧⊞§κ⁰Ｌε⊞ει⊞ζκ»≔ζη»Ｆη⊞§κ⁰¦⁰≔ＥηＬ⊟ιζＩ§η⌕ζ⌊ζ

[Try it online!][TIO-l4ssmyb0] Link is to verbose version of code. Explanation:

    ≔⟦⟦⟦⟧⟦⟧⟧⟧η

Start with no mushrooms placed.

    ＦＥθ÷×÷⊖Ｘ⁴ι²Ｘ²κＸ²ι

For each mushroom, calculate a bitmask corresponding to the space that it occupies. Edit: Saved 3 bytes by using my golf to @Arnauld's formula.

    ¿ι«

If this mushroom occupies space, then:

    ≔⟦⟧ζ

Start collecting placements that include this mushroom.

    Ｆη«

Loop over the existing placements.

    ≔§κ¹ε

Get the bitmasks for the heights as this gets used a lot.

    ＦΦＬε¬＆§ελι

Find whether this mushroom can be placed at any existing height.

    ⊞ζ⟦⁺§κ⁰⟦λ⟧Ｅε⎇⁼νλ⁺μιμ⟧

Generate a placement for putting this mushroom at that height.

    ⊞§κ⁰Ｌε⊞ει⊞ζκ

Generate a placement for putting this mushroom at a new height.

    »≔ζη

Save the new placements as the current placements.

    »Ｆη⊞§κ⁰¦⁰

If this mushroom occupies no space then place it at zero height.

    ≔ＥηＬ⊟ιζ

Get the heights of all of the placements.

    Ｉ§η⌕ζ⌊ζ

Output one of the placements with minimal height.

[Charcoal]: https://github.com/somebody1234/Charcoal
[TIO-l4ssmyb0]: https://tio.run/##bVHBbsIwDD3DV@ToSJkEHTdObAwJaUw9cItyqKihEWkKTQpbJ769c9q1oGmKEsXx83v2yy5Lyl2RmKZZOKcPFqSUSjCpFJ0Zn4/3Rclgk5zgLNja@qW@6BRhq3N0cI@XuCsxR@sxhbi4YgkzwTTngkW0u5dIsCN/jCjPOdN7Bpqz7/Go74CUa1IetdJZm@pzC7@2KX7CUbApUWGAdbiVNp5o39EefAZIyY/Cw4v2V@1wYdOhFAUzvBePK5dBTfPGpnKP7BOCSKPoDLNTzRZLm5Rf8HauEuPAdixtWR7YBMs5V20/LekfrqGvOwJD2RDVwR2KboMPdfcBtzEah6w34z/yCcF@i0K32SAXFycIg3aGxqW2Hl4T5wcCgq40mUNaG211XuVQB2P4vGmkjERYE1oz2s90nyrVPF3MDw "Charcoal – Try It Online"