I edit the quesion according to the comment of professor Edgar
Is it reasonable to suggest to MO to extend the bountiess period from 7 days to 3 or 4 weeks? Can mo be an exception, according to SE policies?
Although the question asks about bounty period, I guess that the most important thing for the OP is whether the bounty brings additional attention and increases chance of getting the question answered. I wonder whether this could be done without changes in the bounty system and bounty period.
What about the following proposal? (If there is need of more detailed discussion, I can post it as a separate question. But I thought that posting it here first to get at least some feedback of the MO community would be reasonable.)
It seems that the OP has mainly the following situation in mind: A user offers a bounty for some question, but they do not receive the answer they were hoping for. Although they might continue adding new bounties indefinitely, they would rather quickly run out of reputation points. Are there other possibilities to attract attention to such questions?
Proposal. What about creating a separate place for this purpose, either in the form of a chat room or as a separate thread here on meta? Here such questions could be collected. That would be beneficial to the users who offered the bounty - in that they would get additional attention for the question. And it would be also good place for users who are looking for interesting questions.
It would be not the first thread collecting interesting questions on this meta: Interesting (and not sufficiently answered) questions on math.SE But perhaps chat room would be less intrusive than a separate thread. Whether the chosen implementation would be to create a chat room or a thread on meta, I do not think it would fill too quickly if the condition for posting would be that at least one bounty has already been placed on the post. (This condition would also prevent users from misusing this by posting there all questions they have.)
I have made a similar proposal on Mathematics site. It seems that the suggestion was met with some enthusiasm judging from the upvotes on meta. (You can also have a look at the rules proposed there - the conditions which the question should meet before it is posted there.) However, the chat room created for this purpose is so far almost unused. (But I would wait a bit before calling this a complete failure. Maybe I was also at fault for not promoting the idea among math.SE users enough. Since there is an overlap between userbase of the two sites, perhaps my post can as a side effect also help to promote that chat room, but the main purpose of this post is to find out what MO community thinks about something similar here.) Considering that MO and Math.SE are rather different (if we compare the size of userbase, activity on meta, rate of asking questions, etc.), it is possible that it could work on MathOverflow even if it fails on Math.SE (and vice versa).
I will add a few comments about the questions posted by the OP - which are unrelated to the above proposal.
Is bounty period sufficient? Here we should first ask: sufficient for what. The purpose of the bounties is partly to incentivize answerers, but mainly to bring attention to the question. So efficiency of the bounties can be measured based on the number of additional views. I guess users skilled with Data Explorer have certainly created various queries examining this, see for example here.
The additional attention depends also on the size of site - mainly on the number of users. And also on the number of bounties. (It is difference whether the additional attention is divided between 10 featured question and 100 featured question. So from this viewpoint making the period longer might increase the number of featured question and be actually counterproductive.)
Can bounty period be changed? The important question here is whether SE team adjusts such settings individually per site or only for the whole network. Although I have no official source of this, it seems that things related to bounties are network-wide settings. See, for example, here, here and here.
In a related thread, it was pointed out that
it seems that some users would consider having somewhere complete list of past bounties a good idea
While it does not do much for extending the bounty period (including very prominent advertising, and the reputation award for people who answer), it is possible to access the list of all bounties on the Stack Exchange Data Explorer (with some bugs for pre-migration bounties like this one, which behave pretty wonkily anyway). The relevant query is this one: