Charcoal, 115 112112 102 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≔⟦⟦⟦⟧⟦⟧⟧⟧ηＦＥθ÷×÷⊖Ｘ⁴ι²Ｘ²κＸ²ι«≔⟦⟧ζＦη«≔§κ¹εＦΦＬε¬＆§ελι⊞ζ⟦⁺§κ⁰⟦λ⟧Ｅε⎇⁼νλ⁺μιμ⟧⊞§κ⁰Ｌε⊞ει⊞ζκ»≔ζη»≔ＥηＬ⊟ιζＩ§η⌕ζ⌊ζ
Try it online!Try it online! Link is to verbose version of code. Explanation:
≔⟦⟦⟦⟧⟦⟧⟧⟧η
Start with no mushrooms placed.
ＦＥθ÷×÷⊖Ｘ⁴ι²Ｘ²κＸ²ιＦＥθ÷×÷⊖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.
Previous less inefficient 112-byte version:
≔⟦⟦⟦⟧⟦⟧⟧⟧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! Link is to verbose version of code. Explanation: Always places zero-width mushrooms at zero height instead of at all possible heights.