The claim that they slept with Rahab is preposterous and virtually unsupported from the biblical text. In fact, the term וישכב that the OP finds so unusual here, to indicate lodging, is not unusual at all. See for example Gen. 28:11
וַיִּפְגַּ֨ע בַּמָּק֜וֹם וַיָּ֤לֶן שָׁם֙ כִּי־בָ֣א הַשֶּׁ֔מֶשׁ
וַיִּקַּח֙ מֵאַבְנֵ֣י הַמָּק֔וֹם וַיָּ֖שֶׂם מְרַֽאֲשֹׁתָ֑יו
וַיִּשְׁכַּ֖ב בַּמָּק֥וֹם הַהֽוּא׃
He reached a certain place and spent the night there, because the sun
was setting. He found a stone there, used it for a pillow, and slept
there for the night. (ISV)
Here we have both Hebrew terms for lodging in the same verse וילן\וישכב. So we see that both terms are equally acceptable for 'lodging' or if you prefer 'settling for the night'. See also 1 Samuel 3 where שכב is used throughout for 'sleeping'; similarly here וישכבו would simply translate into "they went to sleep". As the OP himself points out, וישכבו only has sexual connotations when it is attached to אותה or עמה which mean "with her" or simply "her", but וישכב itself never connotes sleeping with someone! As for the choice of שכב over לין, I think that the former is more specific than the latter. Whereas לין indicates "spending of the night" without specifying how they spent the night, שכב does just that, it clearly designates how they spent the night--through sleeping. Perhaps, the author wishes to convey that the spies had complete trust in their god and were so calm that they even went to sleep in the heart of the enemy's city!
The term באו אלי must not connote a sexual encounter either. Cf 2 Kings 5:22; Isaiah 39:3; Exodus 8:1. Since the text does not make it clear that there was any sexual encounter between the spies and Rahab there is no reason to think that באו אלי implies anything more than 'coming and going'. So I repeat, there is simply no evidence in the bible that the spies slept with Rahab.
It seems to me that this whole speculation is not based on the biblical text itself but on the narrative which relates that they slept in a harlot's house, but of course this is not sufficient proof to support the notion that they actually slept with her. I suspect that it was part of their cunningness and strategy not to raise suspicion among the inhabitants of Jericho, since foreigners were wont to show up at Rahab the harlot's door; this way they ensured a smooth and safe reconnaissance of the city.
Update: It is unlikely, what Soldarnal has suggested, that the author here was trying to portray the spies in a bad light (that they were immoral) for seeking lodging in Rahab's house by using suggestive language. The spies are clearly the heroes of the story here; without them, and the critical information they provided, Joshua couldn't have taken the city. To say that the author was trying to besmirch the heroes of the story for some petty offense is quite far-fetched to say the least.
The term שכב appears in Ruth chapter 3 throughout. Though this word definitely appears in a 'sexual context', there is, however, good reason to believe that there was no sexual intercourse between Boaz and Ruth on that night. So here again, שכב would mean nothing more than 'lie down'. Read here for more on this point. Even if one were to insist that there was a real sexual encounter on that night, this is only due to the "uncovering of his feet" expression that appears in the narrative, without these linguistic hints it would indeed be unjustifiable to assume that there was any sexual encounter between them, and this is especially the case with Josh. 2:1.