Set fee limit for Lightning Close Channel How does a Lightning Network channel initiator set a limit on the fee rate they are willing to accept for the channel close?
And a second question is how can the close fee be negotiated during a collaborative close?
As pointed out the funder of the channel is currently responsible for paying the fees and deciding them. BOLT 02 specifies the update_fee message to be used to signal the fee rate. There it says:

The node responsible for paying the Bitcoin fee SHOULD send update_fee to ensure the current fee rate is sufficient (by a significant margin) for timely processing of the commitment transaction. The node not responsible for paying the Bitcoin fee: MUST NOT send update_fee. A receiving node: if the update_fee is too low for timely processing, OR is unreasonably large: SHOULD fail the channel.

In the case of a collaborative close there is a an algorithm that is part of the closing negotiations:
The closing_signed message contains the follwoing fields
type: 39 (closing_signed)
data:
[channel_id:channel_id]
[u64:fee_satoshis]
[signature:signature]

I quote again:

The sending node: MUST set fee_satoshis less than or equal to the base fee of the final commitment transaction, as calculated in BOLT #3. [...] if the receiver agrees with the fee: SHOULD reply with a closing_signed with the same fee_satoshis value. otherwise: MUST propose a value &quot;strictly between&quot; the received fee_satoshis and its previously-sent fee_satoshis. [...]
The &quot;strictly between&quot; requirement ensures that forward progress is made, even if only by a single satoshi at a time.